Money Capital in the Theory of the Firm: A Preliminary Analysis

{

}

Performing the differentiations and substituting equivalent values from the solution conditions in Equations (4.29) and (4.30) and Equation (4.45), it follows that ^

= M ^ ( e ) / x + a ] ^ + M^(G)/ y + i 8 ] ^ + ( l - X )

(4.50)

This somewhat complex result follows directly from the preceding argument. Note that the last two terms on the right-hand side of Equation (4.49) together describe the effect on the economic value of the equity, V, that results from the changes in r and p, in both cases reductions, as K is increased. Taken together, they represent dV/dK. First, as the interest rate on the debt, r, is reduced, this will reduce the firm's debt interest burden by D dr/dK, and a correspondingly larger residual net income will become available to the equity owners and increase the value of the equity. Second, the reduction in the equity capitalization rate, p, will introduce the equivalent of an increase in the equity earnings stream equal to V dp/dK. Both of these beneficial effects on the value of the equity are shown in the first term on the left-hand side of the final equals sign in Equation (4.45). In that equation they are capitalized at the equity capitalization rate, thereby defining, after taking account of the

4 Production, pricing, investment, and financing

81

signs in the equation, the overall effect on the value of the equity resulting from the reduction of r and p. This resulting overall increase in equity value is seen, by transposition of Equation (4.45), to equal 1 - X. This result, we observe, is also described in the sum of the last two terms on the right-hand side of Equation (4.49), thus making possible the substitution of 1 - X in Equation (4.50). Substituting Equation (4.40) in Equation (4.50) results in ^ = X + (1-X)=1

(4.51)

The analysis in this chapter of structural planning and money capital utilization in the firm suffices for our present restricted purposes. We have looked at the decision problems of the firm as it stands at a point in time and at the determinants of its equilibrium structures. In doing so, we have highlighted, from the paradigmatic perspectives within which the analysis moves, the strands of causation the firm must take into account in deciding its best next move at its structural planning date. But we have not as yet taken significant account of the problem of uncertainty, nor would we wish to be understood to say that the firm is necessarily confronted with stable and well-defined functional relations of the kinds we have examined. We have intended, rather, to bring into focus the lines of causation and general relationships involved, leaving open for the present the possibility that greater or lesser degrees of instability may be assumed to surround those functional relations. We shall return to those highly important questions.

CHAPTER 5

Probability, risk, and economic decisions The question of risk or uncertainty has not yet been brought explicitly into our analysis. It has so far been incorporated only indirectly or in a proxy fashion. In our discussion of the risk exposure of the debt and equity holders in the firm, for example, we examined the impact on their positions of the degree of financial leverage in the firm's financing structure. But recent advances in the theory of the firm have attempted to handle the question of uncertainty in a more direct fashion. It has generally been assumed that the economic variables entering into the analysis could be interpreted as random variables that are describable by subjectively assigned probability distributions. It is risk in that sense that we shall discuss in this chapter.1 We shall be concerned with what has been called variability risk, or risk described by the degree of random disturbance in the economic variables that affect the operating results of the firm. An even more fundamental risk, that of the default, bankruptcy, and possible liquidation of the firm, has not been accorded as extensive a treatment. In focusing on the variability risk, the decision maker has frequently been assumed to choose alternatives on the basis of a utility function defined over the possible profit, economic value, or rate of return that could be generated and the degree of risk involved. The risk, in turn, has been measured by the dispersion parameter of an appropriately defined probability distribution of outcomes. The introduction of a utility function defined over random possible outcomes shifts the theory to what we shall designate as stochastic utility. In the notion of stochastic utility, the probability analysis makes, from the point of view of the received traditions, its ultimate impact on the theory of economic behavior. Probability and probability distributions Consider the information on two investment alternatives contained in Table 5.1. We wish to examine the criteria on which their relative attractiveness to the firm may be assessed. 1

Our present objectives make it neither possible nor necessary to discuss the extensive literature on the application of probability analysis to the decisions of the firm. For a survey of the subject, see Hey (1979), Ford (1983), Weiss (1984), and the extensive references cited there, and for an earlier and important study, see Ozga (1965).

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83

Table 5.1 Proposal B

Proposal A Probability

Cash flow ($000)

Probability

Cash flow ($000)

0.10 0.20 0.40 0.20 0.10

20 30 40 50 60

0.10 0.20 0.40 0.20 0.10

25 30 35 40 45

The data indicate that the firm has reason to believe that the periodic cash flows would range between $20,000 and $60,000 in the case of proposal A and between $25,000 and $45,000 for proposal B. These beliefs, presumably, would be based on an understanding of the firm's past performance in connection with similar investments, the general nature of the economic environment in which the investments would be undertaken, and estimates of the likely trends and instabilities in external economic forces bearing on the firm. The probability data should be read to mean that for proposal A, for example, the investment decision maker believes there is a 10 percent chance that the cash flow will fall as low as $20,000 and a similar 10 percent chance that it will be as high as $60,000. Taking each of the possible cash flow levels, and assigning to each of them in this way a probability of occurrence, the first two columns in Table 5.1 can be read as a probability distribution of the cash flow on Proposal A. In all such probability assignments, the decision maker will allocate a probability magnitude to all the elements of the range of possible outcomes he is prepared to contemplate, and the sum of the assigned probabilities must accordingly be equal to unity. A similar procedure establishes the probability distribution of the cash flow on proposal B, shown in the last two columns of Table 5.1. The cash flows on the projects, it will be noted, are symmetrical around an average level of $40,000 for project A and $35,000 for proposal B. Which, then, is the more attractive investment opportunity? We may suppose that both proposals would involve the same initial capital investment, so that we can concentrate analysis on the shape of the cash inflow prospects in each case. We observe that while proposal A offers the prospect of a higher average cash flow, it also contemplates a wider range within which the actual realized cash flow may fall. It might seem, therefore, that it has associated with it a higher degree of risk. The question arises whether the higher possible cash flow from proposal A would be large enough to offset, in the mind of the decision maker, the higher degree of risk that it involves.

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II The neoclassical tradition

Table 5.2 Computation of expected value and variance of cash flows on proposal A Cash flow

Probability

Xi

Pi

($000)

PiXi

0.10 0.20 0.40 0.20 0.10

20 30 40 50 60

2 6 16 10 6

Xi-E(x) = d,

(dd2

Piidd2

-20 -10 0 10 20

400 100 0 100 400

40 20 0 20 40

2=T20 In Table 5.2, we have set out a computation of the two most important pieces of information contained in the basic data for proposal A. We are interested, first, in what is termed the expected value of the cash flow, or the expected value, sometimes referred to as the mathematical expectation, of the probability distribution of the cash flow and, second, in a measure of the dispersion of the probability distribution. We take as a measure of the latter the so-called variance (or its positive square root, the standard deviation) of the distribution. On the basis of Table 5.2, we can set out the initial definitions relating to the probability analysis. Definition 1 The expected value of a random variable, sometimes referred to as the expected value of the probability distribution of the random variable, is equal to the weighted average value of the possible outcomes, where each such possible outcome is weighted by the probability of its occurrence. Notationally, the expected value is written as follows: (5.1) In the examples contained in Tables 5.1 and 5.2, the random variable x describes the possible cash flows, and xt refers to the /th possible value of that variable. As the sum of the weights in this case, using the probabilities as the weights, is unity, the expected value is simply the sum of the possible outcomes times their probabilities. This is shown at the bottom of the third column of Table 5.2. The expected value may also be written as /JL. We may estimate also the extent to which, on the average, the possible outcomes of the random variable are distributed around, or dispersed away

5 Probability, risk, and economic decisions

85

from, their average or expected value. We are interested for this purpose in the deviation, or difference, of each possible outcome from that expected value. These are shown in the fourth column of Table 5.2. But if we were to take the simple differences as indicated there, the sum of those differences would equal zero, thus diminishing their manipulability. We therefore take the square of the differences, as shown in the fifth column of Table 5.2, and then, as indicated in the final column, we reach the following definitions. Definition 2 The variance of the probability distribution is equal to the weighted average value of the squares of the deviations of the possible outcomes from their expected value, where each such squared deviation is weighted by the probability of its occurrence. That probability is the same as the probability attaching to the outcome whose squared deviation is being considered. Definition 3 The standard deviation of the probability distribution is the positive square root of the variance. The variance and standard deviation are written as = a2x = Sp.-fo - E(x)]2

(5.2)

p,{x,-E(x)]2

(5.3)

Proposal A in Table 5.1 therefore has the following characteristics. Its expected cash flow is $40,000; the variance of the cash flow is $120 (thousands squared); and the standard deviation is $10,950. A further measure of risk that enables comparisons to be made between distributions that are described over different levels and ranges of possible outcomes may be defined. Definition 4 The coefficient of variation, sometimes referred to as the relative standard deviation, is equal to the standard deviation divided by the expected value. The coefficient of variation of proposal A, therefore, is $10.95/40 = 0.27. The coefficient of variation, being a measure of relative risk, is a pure number. Proposal B in Table 5.1 may be analyzed in a manner similar to that adopted in Table 5.2 with respect to proposal A. The comparisons in Table 5.3 between proposals A and B may then be derived. We conclude that proposal A, while it offers a higher expected cash flow, does have associated with it a higher degree of variability risk, as defined by

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II The neoclassical tradition

Table 5.3 Comparative probability data for proposals A and B Statistical characteristic

Proposal A

Proposal B

Expected value Standard deviation Coefficient of variation

$40,000 10,950 0.27

$35,000 5,480 0.16

either the standard deviation of outcomes or the coefficient of variation. The firm could conceivably opt for the relatively safer proposal B, even though by doing so, it would be sacrificing the prospect of a somewhat higher possible cash flow. It is not possible to say immediately which would be the "proper" or the optimal decision. That must depend on the degree of risk aversion with which the firm's decision makers, acting on behalf of the owners of the firm, face their management tasks. The probability distributions we have envisaged so far describe the probabilities attached to certain discrete or uniquely specified possible outcomes, and they are referred to as discrete distributions. In actual fact, the variable x, here representing dollar values of cash flows, may range over a continuous scale of possible values. In that case, the probability distributions will be representable as continuous functions such as depicted in Figure 5.1. In this figure, two continuous probability distributions are shown, approximating the general expected value and variance characteristics of the discrete distributions describing proposals A and B. In the case of a continuous probability distribution, the total area under the curve describing the "probability density function" must equal, or integrate to, unity. We can focus on the area under the continuous probability distribution between any pair of values of the variable described on the horizontal axis and express this as a proportion of the total area under the curve. That proportion or ratio will then describe the probability that the value of the variable will fall within the range between the pair of values indicated. The expected value and variance for the continuous distributions are computed as follows, where x is again a suitably defined random variable: E(x) = f*xp(x)dx

(5.4)

and WaY(x)

= (T2 =

£[Xi_E(x)]2p(x)

dx

(5 5)

In Equations (5.4) and (5.5), p(x) represents the form of the continuous probability density function, and a and b represent the lower and upper limits of the range of the random variable.

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5 Probability, risk, and economic decisions

Proposal B

Proposal A

20

25

30

35

40

45

50

55

60

S000

Figure 5.1

Figure 5.2 The probability distributions of the outcomes may not be of the symmetrical form that has so far been assumed. In Figure 5.2, a "positively skewed" distribution is shown. The cash flow described by such a distribution would hold out the prospect of only a moderate average or expected value, but also the possibility, however small, of very high values. The firm's decision makers could very well be attracted to such a prospect. In that connection, we may anticipate our discussion of the utility functions

88

II The neoclassical tradition

of investors that exhibit varying degrees of risk aversion. The firm may choose between alternative proposals, or between the probability distributions describing the possible outcomes of alternative proposals, on the basis of a utility function defined over the expected values and variances, and possibly also the skewness, of the probability distributions. The form of the utility function may be such that (i) the firm is attracted to higher expected returns, so that the expected-value variable generates a positive marginal utility; (ii) it is averse to variability risk, so that the dispersion parameter, or the standard deviation of returns, generates a negative marginal utility; and (iii) it is attracted to, or derives a positive marginal utility from, positive skewness, such as described in Figure 5.2. Attention has generally been confined to the expected values and variances of probability distributions, and the relevant theory of decision making is widely described as the "mean-variance" theory. Using alternative terminology, the expected value is referred to as the "first moment about the origin" of a distribution, and the standard deviation is referred to as the "second moment about the mean." These statistics are accordingly said to describe the first and second moments of the probability distribution. The measure of skewness is similarly referred to as the third moment. Asset combinations and covariance between asset returns In making asset investments or in choosing a portfolio combination of securities, a decision maker will not generally base his decisions solely on the risk-return characteristics of individual assets or securities. He will be concerned also with the relationships that exist between the return streams generated by the various assets. If, for example, the rates of return on two assets, A and B, were observed to be "positively correlated," then when the return on A increased, the return on B would do likewise. Asset returns that are "negatively correlated" would move over time in opposite directions. This possibility gives rise to the concept of economic diversification. Diversification exists when investable money capital is distributed over assets whose prospective income streams are less than perfectly correlated. In that case, not all the income streams on all the assets rise and fall together, or to the same relative degree, over time. Indeed, if some asset income streams are negatively correlated with others, and they therefore rise when the others fall, the benefits of overall risk reduction are being achieved to a higher degree. For then "what is lost on the swings is being made up on the roundabouts." We approach this important question by examining the concept of the "covariance" between asset returns. Consider the probability distributions of rates of return on two securities, A and B, described in Table 5.4. In this example, the rates of return on the securities, RA and RB, are interpreted as random variables, analogous to the

5 Probability, risk, and economic decisions

89

Table 5.4 Probability distributions of rates of return

Probability

Rate of return on security A (%)

Probability

Rate of return on security B (%)

0.10 0.20 0.40 0.20 0.10

5 6 10 12 15

0.10 0.20 0.30 0.30 0.10

3 4 5 6 8

Table 5.5 Comparative probability data for securities A and B Statistical characteristic

Security A

Security B

Expected value, % Standard deviation, % Coefficient of variation

9.6 2.97 0.31

5.2 1.33 0.26

random variable x in the preceding definitions. An analysis of the mean return and variance characteristics of these two probability distributions reveals the summary data shown in Table 5.5. Here again we have an example of two investment opportunities in which the one promising the higher rate of return, based on the investor's subjective probability distributions, also confronts the investor with the higher degree of risk. Based on both the absolute standard deviation and the coefficient of variation, security A is the more risky opportunity. But it is necessary to take account also of the degree to which covariation occurs, or is likely to occur, over time, between the two rates of return. We therefore examine in more detail the measures of covariation referred to as covariance and correlation. For this purpose, we may imagine that simultaneous observations of the rates of return generated by the two securities are made and recorded in Table 5.6. To determine the degree of the covariation in these rates of return, we set out in Table 5.7 the details of the covariance computation. As Table 5.7 indicates, we are interested now in the simultaneous deviation of the rates of return on securities A and B from their expected values. The data in columns 2 and 4 of the table, given the fact that the expected values of the rates of return have been calculated to be 9.6 and 5.2 percent, respectively, provide the simultaneous deviations in columns 3 and 5. In the final

90

II The neoclassical tradition

Table 5.6 Simultaneously observed rates of return on securities A and B Observation number

Security A (%)

Security B (%)

1 2 3 4 5 6 7 8 9 10

12 6 6 12 10 15 10 10 5 10

5 4 3 4 6 6 8 5 6 5

Table 5.7 Computation of covariance between rates of return on securities A and B Simultaneous observation number

Rate of return on A, RA (%)

1 2 3 4 5 6 7 8 9 10

12 6 6 12 10 15 10 10 5 10

RA-E(RA) = da

Rate of return onfl, RB (%)

RB-E(RB) = db

dadb

2.4 -3.6 -3.6 2.4 0.4 5.4 0.4 0.4 -4.6 0.4

5 4 3 4 6 6 8 5 6 5

-0.2 -1.2 -2.2 -1.2 0.8 0.8 2.8 -0.2 0.8 -0.2

-0.48 4.32 7.92 -2.88 0.32 4.32 1.12 -0.08 -3.68 -0.08

2 = 10.80 column of the table, we record the product of those simultaneous deviations. We are now interested in the weighted average value of those products, or the 4 'expected value" of the product of the simultaneous deviations. We therefore take the weighted average value of the product terms, where each such product term is weighted by the probability of its occurrence. That statistic defines the extent to which the rates of return are fluctuating together. In the present case, each recorded product term has a 10 percent probability of occurring, so that if each product were multiplied by that probability and

5 Probability, risk, and economic decisions

91

the results then added to provide a probability weighted sum, the result would be equal to 1.08. This figure of 1.08 is then the measure of covariance for which we have been seeking. We therefore state: Definition 5 The covariance between two random variables is defined as the weighted average of the products of the simultaneous deviations of each of the random variables from its expected value, where each such product of deviations is weighted by the probability of its occurrence. Notationally, the covariance, in this case the covariance between the rates of return on securities A and B, is written as Cov(/?A, RB) = o ^ = 2 2 P ( * A , RB)[RA-E(RA)][RB-E(RB)]

(5.6)

A B

In this expression, p(RA, RB) represents the joint probability, or the probability of the occurrence together of the RA and RB values that give rise to the deviations contained in the product terms. The expression for the covariance can be written alternatively as o ^ = E[[RA-E(RA)]

[RB-E(RB)]\

(5.7)

The definition and computation of the covariance parallels that of the variance of a random variable. In fact, the variance is a special case of the covariance and can be understood as the covariance of a random variable with itself. In other words, the formula for the covariance given in Equation (5.7) reduces to that for the variance when the variables RA and RB are identical. A further economy of notation describing the variance of the /th random variable in a set of TV variables and the covariance between the /th and the y'th random variable is the following: Var(/?/) = ax Cov(Rh Rj) = (Tij

i = l , . . . ,N

(5.8)

i^j, i,j, = 1, . . . ,N

(5.9)

Correlation between random variables The descriptive statistics we have so far derived from the probability distributions depend on the units in which the original data are measured. If, for example, variables denominated X and Y were, in an appropriate context, measured in feet rather than in inches, the deviations from the expected values, of the kind shown in Table 5.7, would change by a factor of 12, and hence the covariance, CTXY, would also change to a corresponding degree. In

92 II The neoclassical tradition order to obtain a measure of the covariation between two random variables that is unaffected by the units of measurement, we can normalize the deviations by dividing them by the standard deviation of the distribution. The resulting measure, a pure number, is referred to as the correlation coefficient, and the correlation between the ith and the 7th random variables in a set of variables is written as Pij

=- ^ L

i*j,i,j

= 1,.. . ,N

(5.10)

If/ = 7, ptj = 1. We accordingly have: Definition 6 The correlation between two random variables is defined as the covariance between them divided by the product of their standard deviations. Using the expected-value notation of Equation (5.7), the correlation coefficient for the rate-of-return example in Table 5.7 can be written as

It follows that on the basis of the data in Table 5.7 the correlation between the random variables is 3) = 0'21

(512)

The correlation coefficient will always lie between - 1 and 1. In the present case of securities A and B, a moderate, not a high, degree of correlation exists. In our subsequent discussion of security portfolio composition and financial asset market theory, we shall make use of Equation (5.10) to write atj = pijO-idj

i^j,

I , J = 1 , . . . ,N

(5.13)

Much depends, for the possibility of reducing portfolio risk by engaging in asset diversification, on the magnitude of the correlation coefficient between all possible pairs of rates of return on assets, as well as the variances, or the standard deviations, on the individual assets' own rates of return. Because the signs on the standard deviations are positive, the sign of the correlation coefficient follows the sign of the covariance. The lower the correlation, including the possibility of negative value as low as - 1 , the greater the risk reduction that can be achieved by asset diversification.

5 Probability, risk, and economic decisions

93

Linear combinations of random variables: expected values, variances, and covariances In this and the following sections we shall consider the significance of the expected values and variances of combinations of random variables. By doing so, we shall lay the foundation for the traditional theory of the selection of asset portfolios. Imagine that a random observation Xt is made from a set of values of a variable X whose distribution is described by a probability density function having an expected value E(X) and a standard deviation crx. Similarly, we visualize a random observation Yt drawn from a distribution of a variable Y having an expected value and a standard deviation of E(Y) and o>. Imagine now that a new variable 5 is defined as the sum of the simultaneous observations of these X and Y variables. We must interpret S as a random variable because any variable that is a function of one or more random variables is itself a random variable. We now write S=X + Y

(5.14)

It can be shown that E(S) = E(X) + E(Y)

(5.15)

leading to: Proposition 1 The expected value of a sum of random variables is equal to the sum of the expected values of the random variables. Interest frequently attaches to the weighted sum of random variables, where, for example, denoting the weighted sum as 5, S = wlX + w2Y

(5.16)

where w\ and w2 are the weights attached to the random variables. It then follows that E(S) = wxE(X) + w2E(Y)

(5.17)

leading to: Proposition 2 The expected value of a weighted sum of random variables is equal to the weighted sum of the expected values of the random variables.

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II The neoclassical tradition

The computation of the variance of a sum of random variables is a little more complicated. Using the expected-value notation, and writing the variance of a random variable as Var(X) = E[[X- E(X)]2]

(5.18)

the variance of the sum of the random variables X and Y, Var (S), can be written as Var(S) =E[[S-E(S)]2]

(5.19)

Var(S) =E[[(X + Y) - [E(X) + E(Y)]} 2 =E[{[X-E(X)] + [y-£(y)]j ]

(5.20)

or

We now let [X-E(X)] = dx, or the deviation of the X magnitudes from their expected values, and similarly let [Y — E(Y)] = dy. Equation (5.20) can then be written as Var(S)= E[[dx + dy]2]

(5.21)

Recalling the elementary proposition that

we can apply this formula on the right-hand side of Equation (5.21) and write the resulting expression for Yax(S) as Var(S) = E[d2 + d2 + Id^y]

(5.22)

Placing the expectation operator inside the brackets in Equation (5.22) in accordance with Proposition 1, the expression for the variance becomes Var(S) = E(dl) + E( d2) + 2E{d4y)

(5.23)

We recognize that the expectation terms on the right-hand side of Equation (5.23) describe, respectively, the variance of X, the variance of Y, and the covariance between X and Y. The final expression for the variance of the sum of the random variables is therefore Var(S) = Var(X) + Var(y) + 2 Cov(X, Y)

(5.24)

We can therefore state: Proposition 3 The variance of the sum of a pair of random variables is equal to the sum of the variances of the variables plus twice the covariance between them.

5 Probability, risk, and economic decisions 95 The expected value of the sum of the random variables, as defined, for example, in Equation (5.15), takes no account at all, and is in fact quite independent of, whatever covariation might exist between the random variables. But the covariance between the variables, as in Equation (5.24), enters in an important way into the determination of the variance of the sum. Much depends, therefore, on the magnitude of the covariance and, in particular, on the sign of the covariance term. We can see the impact of this in another way. Making use of Equation (5.13), we write Var(S) = Var(X) + Var(y) + 2 pXY(ix
(5.25)

Thus the variance of the sum depends crucially on the sign, as well as the magnitude, of the correlation between the variables. If the correlation between the random variables is negative, or if, that is, P A T < 0 , the variance of the sum will be less than the sum of the variances of the underlying random variables. If the random variables in the sum were the rates of return on assets, the combination of assets whose income streams were negatively correlated would lead to a minimization of the asset portfolio risk. This again is the meaning of economic diversification. Risk reduction can be achieved by combining assets in such a way the correlation between the income streams of given pairs of assets is less than 1. No risk reduction can be accomplished, however, and therefore no economic diversification exists, if investments are made in assets whose income streams are perfectly correlated. The example of asset portfolios motivates our interest in the weighted sums of random variables, where, for example, the weights attached to the variables describe their relative importance in the portfolio. The variance of a weighted sum such as S in Equation (5.16) can be written as 2 Var(5) = J [S-E(S)] ]

{(w{X + w2Y) - [wxE(X) + w2 —E + w2[Y-E(Y)]}2] =E = E[w\d2x + wld2 + 2wlw2dxdy] = w\E(dl) + wiE(dj) + 2wlw2E(dJsdy)

(5.26)

This expression reduces to the following definition of the variance of the weighted sum of random variables: Var(S) = w?Var(X) + w%Vzr(Y) + 2w1w2Cov(X, Y) leading to:

(5.27)

96

II The neoclassical tradition

Proposition 4 The variance of the weighted sum of a pair of random variables is equal to the weighted sum of the variances of the variables (where the weights are the squares of the weights attached to the underlying random variables) plus twice the product of the weights times the covariance between the random variables. The random variables included in the sum or weighted sum may, of course, be extended to any desired number. Let us suppose that a sum of three variables is defined as (5.28) It follows from the foregoing that E(S) = E(X) + E(Y) + E(Z)

(5.29)

Var(S) = Var(X) + Var(7) + Var(Z) + 2 Cov(XY) + 2 Cov(XZ) + 2 Cov(yZ)

(5.30)

and

We must therefore be interested in the covariances between all possible pairs of random variables included in the sum. In the case of a large number of variables, the number of covariance terms that have to be considered mounts rapidly. It is equal to the number of pairs of items that can be defined from the set of, say, n variables included in the sum. Using factorial notation, this is defined as n\ll\{n — 2)\. A more compact way of handling this notational problem, and of visualizing the variances of sums and weighted sums of random variables, is available. It takes up the concept of the variance-covariance matrix, frequently referred to simply as the covariance matrix. Covariance matrix A matrix is a rectangular array of numbers or variables (which may be, and in the present instance will be, a square array) each element of which represents a magnitude or value in which we are for some reason interested. The array of numbers is usually enclosed in square brackets. If we write the variance and the covariance terms in the forms shown in Equations (5.8) and (5.9), Equation (5.30) may be written as O"S = VXX+ &YY+ &ZZ + 2 &XY+2 &XZ+2 VYZ

(5.31)

For convenience of notation, we shall now refer to the variables X, Y, and Z as variables 1,2, and 3, respectively. We can then set out the variance and covariance terms included in the summation in Equation (5.31) in the form

5 Probability, risk, and economic decisions

(731

(T32

033

97

(5.32)

The variance of the sum of random variables, comparing (5.31) and (5.32), will be the summation of all of the terms included in the matrix, the so-called covariance matrix, in (5.32). The covariance matrix has a number of highly important properties. First, it is, of course, square and is in this case said to be of dimension 3 x 3, or three by three. It has three rows and three columns. This is because we must take into account the covariance of each of the variables with each of the other variables. Second, the elements on the principal diagonal of the matrix, here shown as o"n, cr22, and (733, define the variances of the random variables. Including these in the sum therefore takes account of the first three terms in the summations on the right-hand sides of Equations (5.30) and (5.31). Third, the symmetry of the matrix, meaning that for each term on the northeast of the principal diagonal there is a corresponding, and equal, term on the southwest of the diagonal, follows from the fact that the covariance between, say, the second and the third random variables is, of course, the same as the covariance between the third and the second. Fourth, the dimension of the covariance matrix will be equal to the number of random variables in the summation, for example the number of assets included in an investment portfolio. Finally, it follows that an extremely straightforward way exists of visualizing the effect on the variance of a sum of random variables if an additional variable is added to the sum. In the covariance matrix for the simple unweighted sum shown in (5.32), the addition of a fourth variable would expand the variance of the sum by (i) adding a new term, 0-44, to the principal diagonal; (ii) adding a row of covariance terms along the bottom of the matrix, ((T41, 042, CT43); and (iii) adding a corresponding column of covariances to the right-hand side of the matrix. The matrix then assumes dimension 4 x 4 . Covariance matrix of a weighted sum of random variables When the sum of random variables is a weighted sum (as occurs in connection with an asset investment portfolio), the specification of the variance in matrix notation is somewhat more complicated. It becomes even more important, however, for economy of notation and for the visualization of the results. We must now specify, first, what is referred to as the vector of weights attached to the underlying random variables. A vector is a column of elements or numbers enclosed in square brackets, and in the present example the elements would be the weights assigned to each of the variables in the sum.

98

II The neoclassical tradition

Employing, as before, the lowercase wt to refer to the weights, we shall denote the vector by the uppercase W. In the sum of three variables, the vector of weights is written as w2 w3

(5.33)

It is frequently necessary to envisage a so-called row vector of elements, which is understood as the transpose of a column vector. The transpose of the vector in (5.33), notationally referred to as W, appears as W' = [wuw2,w3)

(5.34)

Conceptually, the matrix shown in (5.32) can be visualized as a set of column vectors. It can also be visualized as a set of row vectors. In general, an m x n matrix is one that has m rows and n columns. Alternatively, we can say that a column vector containing m elements is a matrix of dimension m x 1. The transpose, or the corresponding row vector, would be a matrix of dimension 1 x m. The vector in (5.33), for example, can be interpreted as a matrix of dimension 3 x 1 . There exists an extensive and powerful algebra of vectors and matrices, but the main aspect of this that we need to investigate for our present purposes is the operation of multiplication. The quite simple operation of matrix multiplication may appear, at first blush, to be unusually complex. But it can be considerably simplified by examining first the multiplication of vectors. In examining this operation, we should bear in mind that vectors are, as we have seen, a special form of matrix. The steps in the multiplication process that forms the product of vectors will all be reflected in the extension of the ideas to the multiplication of matrices of larger dimension. Let us review, as an example, the calculation of the expected value of a random variable. Equation (5.1) provided the definition of the expected value, and if we now take the probabilities shown there as the weights, the expected value is interpretable, as before, as a weighted sum of possible values or outcomes. The expression for the expected value can then be written as £(*) = 2w/jt/

(5.35)

i

We can now reinterpret this multiplication in terms of vector algebra by visualizing both (i) a vector of possible outcomes, xi9 and (ii) a vector of weights, w/. The expected value of the variable x is then defined as the product of the X-vector of possible values of the variable and the W-vector. Generalizing this procedure will explain the operation in more detail. We shall now use the symbol X to denote a vector of variables (not to be confused with the previous use of x to refer to a single variable). For example, the

5 Probability, risk, and economic decisions 99 variables X, Y, and Z in Equation (5.28) can now be understood to be represented by x\, X2, and JC3, as shown in X=

(5.36)

We also have a vector of weights in (5.33). The vector multiplication now proceeds as follows. We multiply the first element of the X-vector by the first element of the W-vector to obtain VVIJCI. We then multiply the second element of the X-vector by the second element of the W-vector, and so on throughout the range of elements. This provides a set of product terms of the kind indicated. If we then take the sum of those product terms, we have precisely what is described as the product of the two vectors. What we have accomplished, in performing these successive multiplications of X-vector elements by W-vector elements and adding the results, is what is known as the operation of the multiplication of vectors. It is sometimes referred to also as taking the inner product of vectors. Notationally, the operation is written in summary form as W -X. It is also referred to as the dot product, because the dot between the transpose of the W-vector on the left and the X-vector on the right is understood as the sign indicating multiplication. Vectors are said to be "conformable for multiplication" only if they contain the same number of elements. Moreover, it is usual to write the vectors being multiplied together in the form W'-X=[wi, w2, w3]

(5.37)

That is to say, the vector on the left is written as a row vector, or as the transpose of the vector of weights, and that on the right is written as a column vector. Recalling that we have already interpreted a column vector containing m elements as a matrix of dimension m x 1 and its transpose as a matrix of dimension 1 x m, we can now refer to this vector multiplication as the simplest form of matrix multiplication. Taking this view of things, we note that the matrix on the left [the 3-element row vector in (5.37)] contains as many columns as the number of rows contained in the matrix on the right [the 3element column vector in (5.37)]. This fact should be recognized clearly and borne in mind in what follows. It specifies the very important rule that must be satisfied in order to enable us to say that matrices are "conformable for multiplication." When we deal with higher-order matrices, instead of the simple 1 x m and m x 1 matrices we have in the vector multiplication case, we shall see again that matrices can be multiplied together only if this rule is

100

II The neoclassical tradition

satisfied. The matrix on the left must have as many columns as the number of rows in the matrix on the right. By way of final definition, a matrix of dimension 1 x 1 is simply a single number. It is referred to as a scalar. An interesting result follows from the foregoing. When we multiply a 1 x m vector on the left by an m x 1 vector on the right, as we have done in (5.37) in the 3-variable example, we obtain as the result a 1 x 1 matrix, or a scalar. The number of rows in the matrix that gives the result of the multiplication, the so-called product matrix, will be the same as the number of rows in the matrix on the left, and the number of columns in the product matrix will be the same as the number of columns in the matrix on the right. Thus, for example, the multiplication of an m x n matrix by an n x m matrix will provide a product matrix of dimension m x m. We must now expand these ideas to demonstrate the multiplication of matrices of higher order. Recall the covariance matrix shown in (5.32), which we shall refer to in what follows by the symbol V, and the vector of weights shown in (5.33). Interpreting the transpose of the vector of weights as a matrix of dimension 1 x 3, we observe that such a matrix and the covariance matrix are conformable for multiplication. We set them together in the following way: w2, w3]

(722

^23

032

^33

(5.38)

Multiplying the row vector by the first column of the matrix on the right, we obtain, by applying now familiar procedures, a scalar equal to the sum of w\(T\\ + w2o"2i +W3031. If, similarly, we multiplied the row vector on the left by the second column of the covariance matrix, we would obtain another scalar result. We could again obtain a further scalar result by multiplying the row vector by the third column of the covariance matrix. By performing the multiplication envisaged in (5.38), we would obtain a product matrix of dimension 1 x 3 . This would be written in the form W'V= [Zwicrn, E * W 2 , S ^ a ]

i=l,2, 3

(5.39)

In Equation (5.39), we have omitted from the left-hand side, for economy of notation, the dot that was employed in the illustrative example in Equation (5.37). We now take this 1 x 3 matrix and multiply it by the vector of weights, this time placing the vector of weights on the right-hand side. This is necessary because the vector of weights, being interpretable as a 3 x 1 matrix, will then be conformable for multiplication with a matrix on the left of dimension 1 x 3. So then, having the product matrix derived from (5.38) on the left [the 1 x 3 matrix in (5.39)] and the vector of weights on the right (a 3 x 1 matrix), the resulting product matrix will be of dimension 1 x 1, or a scalar.

5 Probability, risk, and economic decisions 101 To obtain this result, we proceed substantially as before. The operation is defined in Equation (5.40). We take the first element of the row matrix on the left and multiply it by the first element of the column vector on the right. This multiplication will provide us with a scalar. We repeat the procedure and multiply the second element of the row matrix on the left by the second element of the vector on the right. This provides a second scalar. Then we obtain another scalar by multiplying the third element of the row matrix by the third element of the vector on the right. We have, then, three scalar products. Adding these products provides a scalar sum, which is the result we have been searching for. It is the product of the 3-element row vector and the 3element column vector. It is an application of the operation of the multiplication of vectors that we illustrated initially. It provides, in fact, the variance of the weighted sum of random variables. The operation can be summarized as follows: i=l,2,3 (5.40) w2 w3 Let us recall what we have done. Employing the notation V to refer to the covariance matrix and W to denote the vector of weights, we have performed a two-stage matrix multiplication that can be summarized in matrix notation as The variance of the weighted sum = W'VW

(5.41)

leading to: Proposition 5 The variance of a weighted sum of random variables is equal to the transpose of the vector of weights multiplied by the covariance matrix multiplied by the vector of weights. We may make a final interpretation of the result achieved by effecting the multiplications in Equations (5.39) and (5.40) leading to the variance of the weighted sum as defined in Equation (5.41). Setting out the full expansion of Equation (5.40), we have ()

(

= W\(T\\ + WiW2CT2\ + W1W3O31 + W2CT22 + W2W3O32 + W^WiCT 13 + W3W2<X23 + ^30*33

(5.42)

This sum may be restated in terms that reflect precisely the variance of the weighted sum defined in Equation (5.27), noting again the symmetry in the covariance terms included in the summation in Equation (5.42). For example, (J2\ is the same as CT\2.

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II The neoclassical tradition

To reinforce the procedures of this important matrix multiplication, let us work through an analogous example. We shall define a 3-element vector and a 3 x 3 matrix as follows (the numbers in the example are used simply for illustrative purposes and have no actual or theoretical meaning or application): "2 W= 3 5

3 7 5" 1 4 6 2 1 3

V=

We wish to calculate the product of W'VW. It follows that

W'V=[2,3,5]

3 7 5' 1 4 6 = [19,31,43] 2

1 3

Referring to the result of this multiplication as matrix M of dimension 1 x 3 , and multiplying it n o w b y the W-vector on the right, w e have

MW=[\9, 31,43]

= [346]

Thus, the scalar 346 is the result of the multiplication W'VW. Variance of a weighted sum of random variables Given that the variance of a weighted sum of three random variables can be expressed as the matrix product WVW, we can extend the argument and envisage the variance of a weighted sum of n random variables. The magnitude n may be, for example, the number of financial assets included in an investment portfolio, or the number of assets in an opportunity set from which an optimal portfolio is selected. We thus envisage a vector of weights defined as Wf = [w\, vv2, • • • , wh - - • , Wn]. Similarly, we envisage an n x n covariance matrix. The latter takes the general form 022

V=

(5.43)

Such a covariance matrix may be multiplied by the transpose of the vector of weights on the left (that is, premultiplied), and by the vector of weights on the right (that is, postmultiplied) to provide the variance of the weighted sum. If we envisage the set of random variables as describing the rates of return on

5 Probability, risk, and economic decisions

103

the securities in a portfolio, we could summarize as follows the effect on the variance of the portfolio rate of return that results from adding a new asset to the portfolio. First, the addition of the asset would introduce a new variance term to the principal diagonal of the covariance matrix, which would then be of dimension (n+ 1) x (n+ 1). Second, we should add a row of covariance terms along the bottom of the covariance matrix and a corresponding column of covariance terms on the right-hand side of that matrix. Third, we should also change the vector of weights that would become an n + 1-element vector. If, moreover, the weights were taken to be the proportion of the total portfolio represented by the different assets (making it necessary for the weights to sum to unity), the introduction of a new asset would necessarily change the weights given to all of the previously existing assets in the portfolio. Having calculated the variance of the weighted sum of random variables, we can summarize the results for the n-variable case as follows, where S again refers to the weighted sum: Var(S) = ^wTVarto) + 2 2 2 WiWjCovfaxj)

(5.44)

i=\j>i

Employing alternative notation, this can also be written as (5.45) /

j

Again, recalling the definition in Equation (5.13), the final term in Equation (5.44) can be replaced by the term n

n

22 This development will have application in the discussion of the theory of financial asset markets and the firm's cost of money capital. We conclude this chapter by considering in a preliminary fashion the economic opportunities provided by asset combinations in the light of the different degrees of correlation that might exist between individual asset returns. Application of probability and risk: two-asset portfolio opportunity locus With the foregoing definitions and computational procedures in hand, we may return to the two-security example we considered earlier in this chapter. We envisage now the possibility of investing in two assets, A and B, where the rates of return on the assets are described by probability distributions with

104

II The neoclassical tradition

E(R)

E{RR

o(RA)

o(RB)

o(R)

Figure 5.3 expected values and variances of E(RA), E(RB), Var(/?A), and Var(/?#), respectively. We shall refer to the weights assigned to each of the assets as the proportion of the portfolio accounted for by each asset, designated as wA and wB. Denoting the rate of return on the portfolio as Rp, where the subscript here focuses our attention on the portfolio outcome, the expected rate of return on the portfolio will be interpretable as the weighted sum of the rates of return on the individual assets comprising the portfolio. It follows that the expected value and the variance of the rate of return on the portfolio can be defined as (5.46) and ) + 2wAwBCov(RA,

RB)

(5.47)

The covariance term in Equation (5.47) may be written alternatively as Cow(RA, RB) = pAB(rAcrB

(5.48)

Consider now the coordinates of the asset A and B investment opportunities in the risk-return plane in Figure 5.3. We there refer to the risk of an asset as indicated by the standard deviation of its rate of return. Let it be supposed that the asset returns are perfectly correlated, or that pAB = 1. Then if Equation (5.48) is substituted into Equation (5.47), it follows that the variance of return on the portfolio can be written as

5 Probability, risk, and economic decisions 2

Var(/?p) = [wA(jA + wBaB]

105 (5.49)

where the standard deviations of the rates of return on the assets are now written as aA and aB. This follows from the fact that as the correlation between asset returns is assumed to be unity, the variable pAB falls out of the expression for the portfolio variance. By expanding the right-hand side of Equation (5.49), the result in Equation (5.47) [after the substitution of Equation (5.48)] is obtained. Equation (5.49) implies that a(Rp) = wAaA + wBaB

(5.50)

In this special case of perfectly correlated asset returns, the standard deviation of the portfolio rate of return is equal to the weighted sum of the standard deviations of the rates of return on the assets contained in the portfolio. We can now trace out, in the light of these definitions, the effect on what we shall call the portfolio investment opportunity frontier, or the portfolio opportunity locus, as this is determined by different possible values of the correlation coefficient p^. This portfolio opportunity locus, such as the locus AB in Figure 5.3, may or may not be linear. It describes the rate at which an investor, by varying the proportionate combination of securities A and B in the portfolio, can vary both the expected return on the portfolio and the degree of risk as measured by the standard deviation of the portfolio rate of return. If, for example, only security A were held, the portfolio risk-return characteristics, E(RP) and (r(Rp), would be described by the risk-return characteristics of security A, or by the coordinate point A in Figure 5.3. By introducing security B to the portfolio and investing in a weighted combination of the two assets, the resulting portfolio return and risk will be described by a different point in the locus AB. The slope of this locus therefore describes the rate at which the portfolio expected rate of return can be increased in compensation for undertaking a higher degree of portfolio risk. We note that in the two-asset case the value of wB may be written as 1 — wA. A change in the proportions in which the assets are combined in the portfolio may then be reflected simply in a variation in the magnitude of wA. We may substitute wB = 1 — wA in Equation (5.46) and differentiate with respect to wA, thus observing the effect on the portfolio return of increasing the relative weight of asset A: (5.51) Similarly, differentiation of Equation (5.50) with respect to wA implies da(Rp)

(5.52)

106 II The neoclassical tradition Dividing Equation (5.51) by Equation (5.52) provides the explicit relation between E(RP) and oiRp) that we have anticipated: dE(Rp)_E(RA)-E(RB) CTA - crB

(5.53)

Equation (5.53) then describes the rate at which the portfolio rate of return can be increased by increasing the degree of risk contained within it. This rate of change describes, as we have seen, the portfolio opportunity locus. In the present case, which has been derived from the assumption that the rates of return on the underlying assets are perfectly correlated, the opportunity locus is linear. This follows from the fact that the magnitude of all of the terms on the right-hand side of Equation (5.53) are given and fixed once the forms of the probability distributions of the separate asset returns are specified. If the value of pAB in the preceding derivation were less than unity, or if the asset returns were less than perfectly correlated, it would follow from Equation (5.47) [after the substitution, as before, of Equation (5.48)] that the portfolio variance, and accordingly its standard deviation of rate of return, would be lower than for the case of p/& — 1. We can consider the case where pAB= — 1. It follows from Equations (5.47) through (5.50) that o-(Rp) = wAaA - wBaB

(5.54)

or, recalling that wA + wB = 1, o-(Rp) = wAaA - (1 - wA)o-B

(5.55)

In this case a zero-risk portfolio may be achieved, given the assumptions regarding the risk-return characteristics of the assets and the assumed covariance between their rates of return, if Equation (5.55) is set equal to zero and solved for wA. The result indicates that this is achieved by setting WA = o-B/(aA + o-B)

(5.56)

Generating a similar result for wB by taking wB = 1 - wA, it can be shown that the zero-risk portfolio will be achieved when the assets are combined in such proportions that wA/wB = crB/crA, or when the ratio of their weights is equal to the inverse of the ratio of the standard deviations of their rates of return. When the correlation between the asset returns, therefore, is equal to — 1, the portfolio opportunity locus in Figure 5.3 degenerates to the two-segment linear locus AFB, where the point F reflects the zero-risk combination of assets. The prospect of a zero-risk portfolio is possible because, under the assumptions stated, variation in the rate of return on one of the assets is precisely offset by (is negatively correlated with) the simultaneous variation in the rate of return on the other asset. In actual fact, interest usually attaches to the possibility that the correlation

5 Probability, risk, and economic decisions 107 coefficient will lie between — 1 and 1, such as in the example discussed in Tables 5.4 through 5.7. The portfolio opportunity locus will be of the kind shown in Figure 5.3 as the nonlinear locus AB lying between those already derived for the extreme cases of perfectly positive and perfectly negative correlation. The correlation between the assets may in certain cases be zero. In such an event, it will be possible to determine the point on the nonlinear locus such as AB that provides the minimum-risk portfolio. Given that the correlation coefficient is equal to zero, it follows from Equation (5.47) [after the substitution, as before, of Equation 5.48)] that the variance of the portfolio return will be equal to the weighted sum of the variances of the individual assets. Writing this variance as *2P = \fi(r2A + {l-WA)2
(5.57)

we may take the derivative with respect to wA and set it equal to zero to find the minimum-risk portfolio. It can be shown that this will be achieved when the assets are combined in such a proportion that wA/wB = cri(r\. In this case, the ratio of the asset weights will equal the inverse of the ratio of the variances of their rates of return rather than the inverse of the ratio of their standard deviations that we observed in the case where p^s — ~~ 1 -2 2

The extensive literature in the area discussed in this section stems from the seminal work of Markowitz (1952, 1959), Sharpe (1964, 1970), and Lintner (1965). See also Tobin (1965), Mossin (1973), Arrow (1971), Hicks (1967), Vickers (1978), and Ford (1983).

CHAPTER 6

Utility, uncertainty, and the theory of choice The objective of economic theory, it is frequently suggested, should be that of explaining ' 'human behaviour as a relationship between ends and scarce means which have alternative uses" (Robbins, 1935, p. 16). In this conception of things, the problem of allocation comes sharply into prominence. It has led to the view that economic theory is concerned essentially with the logic of choice. Difficulties inhere in the constriction of economics to such a narrow focus, but it has given rise to extensive investigations into the criteria of choice or action. Classically, the criterion on the production side of the economy has been that of profit maximization or, as we have seen, the maximization of economic value. On the consumption side, the criterion has generally been the maximization of utility or satisfaction, and a considerable body of theory has developed around the notion of the consumer's utility function. This provides a linkage with our present analysis, pointing to the possibility that decision makers' utility functions may be defined, not only over assumedly known conditions or entities, but over variables whose values can be specified only probabilistically. We raise, therefore, the possibility of what we refer to as the theory of stochastic utility, or of utility functions defined over stochastic arguments. In this chapter we shall examine a number of basic and related propositions that project analogies onto the theory of the firm.

Utility function It is not necessary to assume that utilities, or degrees of satisfaction, are measurable in cardinal units. The theory in its developed form has become a theory of ordinal, rather than cardinal, utility (see Stigler, 1950; Shackle, 1967, Chs. 7, 8; Page, 1968; Bernoulli, 1968; Friedman and Savage, 1968; von Neumann and Morgenstern, 1953, 1968). An individual is assumed for this purpose to be able to order all possible pairs of objects of choice, say commodities X and Y, in such a way as to state that he prefers X to Y, or Y to X, or that he is indifferent between them. Only one such statement, of course, is assumed to be applicable to each given pair of objects. Moreover, the statement that the individual prefers X to Y (ranks X ahead of, or superior to, Y in 108

6 Utility, uncertainty, and theory of choice

109

his "preference ordering") and Y to Z implies that he prefers X to Z. In formal terms, the first statement will reappear as the axiom of completeness of ordering and the second as an instance of the axiom of transitivity. These postulates, together with other assumptions we shall explore more fully, lead to the implication that the individual can describe his preference ordering by an ordinal utility function. This means that he can assign real numbers or specifications of magnitude to each of the objects of choice but that those numbers do not represent absolute magnitudes of the amount of satisfaction the objects of choice provide or are expected to provide. The numbers indicate merely an order of ranking. It follows that if an ordinal utility function is taken to represent a preference ordering, any monotonically increasing transformation of that function will also represent the ordering. The ordinal utility function, that is to say, is not unique. Or in other terms, it is unique only up to a monotonically increasing transformation. By this term, the following is implied. We can suppose that U =f(X, Y) describes a utility function defined over objects of choice consisting of quantities of commodities X and Y. Imagine that another utility index, or utility function, T=g(U) is defined by applying a monotonically increasing transformation to the function U. The function g(U) is a monotonically increasing transformation of U if g(U\)> g(Uo) whenever U\>UQ. Then the utility function T also represents the preference ordering. It follows that if, working with the original utility function, a number of 60 is assigned to one combination of X and Y and a number of 30 is assigned to a second such combination, it is not possible to say that the first combination provides twice as much utility as the second. In order to be able to say, moreover, that a preference ordering can be represented by an ordinal utility function, or that it is, in formal terms, a representable preference ordering, a number of characteristics of the ordering must be present. First, in addition to the notions of completeness and transitivity, the set of possible objects of choice must be "connected." That is to say, if A\ and A2 are combinations of commodities in the opportunity set available to the individual, it must be possible to proceed from Ai to A 2 along a continuous path of such combinations. This important assumption of continuity will reappear in a significant guise in the theory of preference orderings over stochastic outcomes. Second, we can consider all the combinations of commodities in the closed opportunity set that are at least as well liked as a given combination, say Ai, and also the set of combinations that are preferred to A\. This means that if we envisaged an infinite sequence of commodity combinations in, say, the set of combinations preferred to A\ and such a sequence converged to a limiting combination of Ao, that limiting combination Ao would also be preferred to A\. Let us consider an ordinal utility function

110

II The neoclassical tradition

U=f(X,Y)

(6.1)

It may appear that we could envisage the increment of utility that would result from adding a marginal increment of, say, commodity X to the consumption combination. It might be thought that in this way, assuming the utility function possesses appropriate differentiable properties, we could define the partial derivative of the utility function with respect to X as the marginal utility of X. But difficulties attach to such an interpretation of the ordinal and nonunique utility function we have just described. We shall be interested, however, in the ratio of the marginal utilities of the commodities, and this, which is quite properly reflected in the ratio of the partial derivatives of the utility function, has valid and significant meaning. Such a ratio remains defined over a monotonically increasing transformation of the original utility function. We may take, for example, the function U = aX + bY and a monotonic transformation of it, T=U2. The function T can then be written T=a2X2 + b2Y2 + labXY. The ratio of the partial derivatives of the function U can be shown to be equal to alb. When the similar ratio of partial derivatives is calculated from the function T, it is found to be equal once again to alb. This ratio of partial derivatives will be referred to as the marginal rate of substitution between the commodities, or between whatever happens to be defined as the objects of choice that form the arguments of the utility function. At this point, the sole property of the consumer utility function in which we are interested is the manner in which it gives rise to a commodity indifference map. This map is made up of a family of indifference curves in the commodity space such that each curve in the family describes a set of commodity combinations that afford the individual the same level of utility. In that sense, he is indifferent between them. They rank equally in his representable preference ordering. Members of such a family of indifference curves are shown in Figure 6.1. The marginal rate of substitution between the arguments in the utility function, in this case commodities, can be developed in the following manner. Consider the utility function in Equation (6.1). We are interested in the change in the value of the function that would result from a simultaneous change in the magnitude of the arguments X and Y. We specify this by writing the total differential of the function: dU = ^dX

+ ^dY

(6.2)

Notationally, dU/dX and dU/dY in Equation (6.2) may be referred to as/* and fy, respectively. The overall change in the value of the utility function is defined as the rate of change with respect to commodity X, or the partial derivative with respect to X, multiplied by the amount of the change in X, that

6 Utility, uncertainty, and theory of choice

111

Figure 6.1 is, dX, plus the rate of change with respect to Y multiplied by the amount of the change in Y. We consider now a simultaneous change in the arguments X and Y that leaves the individual on the same indifference curve, or at the same level of utility. In that case we can set Equation (6.2) equal to zero. It follows by transposition that dY dX'

fy

(6.3)

The derivative dYldX shown in Equation (6.3) describes the slope of the indifference curve in Figure 6.1. We conclude, therefore, that the slope of the indifference curve is defined as the negative of the ratio of the partial derivatives of the utility function, or as the ratio of the marginal utilities. It defines the marginal rate of substitution between the commodities X and Y, or the rate at which, in order to remain at the same level of utility, the individual is willing to substitute commodity X for commodity Y. The consumer's utility maximization problem may be solved against an income constraint that restricts him to an attainable commodity combination, or to an attainable commodity set such as the area OAB in Figure 6.1. At the optimum-commodity combination point, or at the point of constrained utility maximization, the slope of the indifference curve will be equal to the slope of the relative commodity price line AB that forms the boundary of the attainable

112

II The neoclassical tradition

set in the commodity space. The location of that boundary is set by the total income or resources available for consumption. The fact that the budget constraint is operative ensures that the two equal slopes meet in a tangency. This tangency condition is shown in Figure 6.1 at point D. The individual will consume X* of commodity X and 7* of commodity Y. An analogous tangency condition will occur in the case of a portfolio investor's optimization over a stochastic utility function. Utility and probability: investor's risk-aversion utility function Consider now the investment opportunities and prospects confronting an investor who approaches the financial asset market at the beginning of time period t with a specified total wealth endowment of Wt. We can conceive of the rate of return, Rt9 that might be earned on that wealth portfolio during period t. In the manner of Chapter 5, we can visualize that rate of return as a random variable describable by a subjectively assigned probability distribution, such as is shown in Figure 6.2, with expected value E(Rt) and standard deviation (jiRt). The probability distribution of the rate of return might or might not be the approximation to the normal distribution depicted in Figure 6.2. If the distribution is assumed to be normal, the third moment, or the measure of skewness, will be equal to zero, and the investor's attention will be focused on the first two moments, the expected value and the standard deviation. This is the usual form of distribution assumed in the traditional theory, which is accordingly referred to as the mean-variance theory of investor's portfolio optimization. Consider now the potential accumulated value of the investor's wealth at the end of time period t: Wt+l = (l+Rt)Wt

(6.4)

As the rate of return on the portfolio is a random variable, we must interpret the end-of-period wealth, Wt+\, as a random variable also. We are interested, then, in the attractiveness to the investor of, or the different possible levels of utility provided by, the possible values of Wt+i. It follows that we can focus on the properties of the random rate of return that generates the stochastic nature of the wealth variable. For purposes of exposition we shall assume an ordinal utility function defined over this random rate of return, and for concreteness we shall assume it to be a second-degree utility function of the form U(R) = aR-bR2

(6.5)

This function specifies the level of utility that the individual would attach to different outcome values of the rate-of-return variable. As utility is now a function of a random variable, however, utility itself must be interpreted as a

6 Utility, uncertainty, and theory of choice

113

Figure 6.2

random variable. This implies that Equation (6.5) does not possess differentiate properties. No meaning attaches to the derivative of a random variable with respect to another random variable. No meaning would attach, therefore, to the notion that the marginal utility of the rate of return might be described by the derivative of the utility function with respect to the return variable. In order to address the concept of marginal utility, the utility function in Equation (6.5) must be transformed into another function that does have differentiate properties. We accomplish this transformation by taking the expected values of both sides of Equation (6.5). The development in Chapter 5 implies that E[U(R)] = aE(R)-bE(R2)

(6.6)

The second term on the right-hand side of Equation (6.6), the expected value of the square of the underlying random variable, can be shown to be equal to the sum of (i) the square of the expected value and (ii) the variance of the probability distribution of the random variable. That is, This can be confirmed from an inspection of the probability distribution in Figure 6.2. Consider the possible outcome recorded in Figure 6.2 as Rt. This deviates from the expected value by the amount dt. It can therefore be written as Rt = E(R) + dt. Taking the square of this magnitude provides (6.8)

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II The neoclassical tradition

Equation (6.6), however, requires us to evaluate the expected value of R2. We therefore take the expected value of Equation (6.8). In doing so, we interpret the expected values of the terms on the right-hand side of the equation in the following manner. The expected value of the first term, the expected value of the square of the expected value of the distribution, is simply the square of the distribution's expected value. This is because that expected value is a datum, or a constant, once the form of the underlying probability distribution has been specified as in Figure 6.2. The expected value of the second term, or E(dj), is recognizable as the variance of the probability distribution. Finally, the expected value of the third term, or 2E(R)E(di), is equal to zero because the last element in this expression is simply the weighted sum of the deviations of the possible values of the variable from their average, or expected, value weighted by their associated probabilities. We therefore write the expected value of Equation (6.8) in the form of Equation (6.7) as required. Taking Equation (6.7) and substituting into Equation (6.6) and writing the expected value of the probability distribution of the rate of return as /UL and the variance as a2, the expected utility function can be written as E(U) = aim- b\x2 - be2

(6.9)

It is important to note what has been accomplished by this transformation. We have taken a second-degree utility function defined over the random variable rate of return and have transformed that into an expected utility function defined over the first two moments of the probability distribution of the underlying random variable. If there had been reason to believe that the probability distribution of the rate of return on the portfolio possessed a positive measure of skewness, and if, furthermore, the investor was attracted to the possibility of the high rate of return that such a positive skewness indicated, we could have begun with a third-degree utility function defined over the random variable. Such a third-degree stochastic utility function would be transformed into an expected utility function defined over the first three moments of the distribution of the underlying random variable. Generalizing from this result, an nth-degree stochastic utility function can be transformed into an expected utility function defined over the first n moments of the probability distribution of the underlying random variable over which the stochastic utility function is defined. If, then, the investor is optimizing over a second-degree, or quadratic, utility function, there is no point in examining the third or higher moments of the relevant probability distribution. Similarly, if the probability distribution is assumed to be a two-parameter distribution, such as the widely assumed normal distribution of economic and financial data, there is no point in assuming that the utility function is of higher than second degree. This provides the basis for the mean-variance theory of portfolio selection and optimization.

6 Utility, uncertainty, and theory of choice

115

Given the expected utility function of Equation (6.9) and recognizing its differentiable properties, we can envisage the marginal utility of expected return by taking the partial derivative with respect to the expected-return variable:

*fpi = .-2^

(6.10)

Similarly, the marginal utility of risk, or the partial derivative with respect to or, provides

m

(6.11)

da

These results indicate that the marginal utility of risk, as indicated in Equation (6.11), is everywhere negative. We say, therefore, that an individual optimizing against a utility function of this form is a consistent risk averter, or that the utility function is a consistent risk-aversion function. Other functional forms and measures of risk aversion have been defined in the extensive literature on utility functions. The illustrative form with which we have worked is frequently referred to as a Tobin function (Tobin, 1958). It would be possible to adopt an exponential utility function of the form U = a-ce~bR

a^0,c,b>0

(6.12)

which transforms into an expected utility function (see Ford, 1983, pp. 23-4, 191-92) defined as E(U) = a-ce~b^R-(b/2)(rR]

(6.13)

An interesting difference exists between the Tobin function and that in Equation (6.12). In the former case, it appears from Equation (6.10) that the marginal utility of expected return will become negative for values of expected rate of return greater than allb. In the latter case, the marginal utility of expected return is everywhere positive, and the E(U) function has an upper bound of a. It is not necessary for analytical purposes, therefore, to place a limit on /UL to guarantee that the marginal utility is positive as in the case of the quadratic utility function. If it can be assumed that the marginal utility of expected return is positive, and if the utility function is a consistent risk-aversion utility function where the marginal utility of risk is everywhere negative, as in Equation (6.11), the iso-utility or indifference contours in the risk-return plane will be positively inclined as in Figure 6.3. As in the analysis of the consumer utility function, the slope of the indifference curve is defined by the negative of the ratio of the marginal utilities or the partial derivatives of the function with respect to

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II The neoclassical tradition

o(R)

Figure 6.3 its arguments. Making use of Equations (6.10) and (6.11), this slope is defined as d/JL _

da

2bcr

a — 2b JX

(6.14)

In Figure 6.3, the arrow indicates the direction of increasing utility. The convexity of the indifference curves can be established by taking the second derivative d2/ui/dcr2, which may be shown to be positive in the present example. This analysis provides the basis for a choice between alternative portfolios depending on the probability distributions of their rates of return. If the expected values and standard deviations of portfolio rates of return are substituted into the expected utility function, the relative attractiveness of portfolios can be compared and evaluated. That portfolio will be the most attractive that offers the investor the prospect of achieving the highest attainable indifference curve. It is for this reason that decisions under uncertainty, or what we are here considering as probabilistically reducible risk, have been referred to as 4 'choices among probability distributions" (Mossin, 1973; Hirshleifer and Riley, 1979). Different possible degrees of risk aversion are exhibited in Figure 6.4. The indifference curves A and B shown there are derived from different utility functions, and each is a member of a different indifference map. It is imagined that the investor occupies the position marked P, holding a portfolio that promises an expected return of E{RP) and has a standard deviation of return of cr{Rp). Moving along the risk axis to the right of P, it is readily seen that a

6 Utility, uncertainty, and theory of choice

117

E(R)=L B

| Aa o(R)

Figure 6.4 higher increment of expected return would be required if the investor were optimizing over a utility function that generated indifference curve B than if his utility function had generated indifference curve A. The utility function that gives rise to indifference curve B would therefore exhibit the greater degree of risk aversion. When it is said that an investor is risk averse, this implies that his attitude to risk is such that he will reject what has been called a fair gamble. This refers to a situation in which the expected value of the outcomes of a risky prospect is equal to the amount that would have to be paid for the opportunity to acquire that prospect. For example, a lottery that offered two alternative outcomes of two dollars and zero dollars, where each outcome had a 0.50 probability of occurring, would have an expected value of one dollar. A "fair gamble" would be the exchange of the lottery prospect for its expected value of one dollar. The investor would stand to realize a net gain of one dollar if the two-dollar outcome occurred, or a net loss of one dollar in the event of the alternative outcome. But a risk-averse investor would reject such an exchange because the disutility to him of the loss would be greater than the utility he would obtain from an equivalent gain. His utility function, such as that assumed in Equation (6.5), would therefore be concave. An alternative measure of risk aversion has been constructed by focusing on what has been called the Absolute Risk-Aversion (ARA) or the Relative Risk-Aversion (RRA) properties of the utility function. A decreasing absolute risk aversion is said to exist if an investor is prepared to increase the amount of his wealth invested in risky assets when the total amount of his wealth increases. The opposite behavior would exhibit increasing absolute risk aver-

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II The neoclassical tradition

sion. Take, for example, a utility function defined over total wealth of the form U(W) = W-cW2

(6.15)

and employ the notation U'{W) and U'\W) to refer to the first and second derivatives of that function as defined over actual wealth outcomes. Then (see Elton and Gruber, 1984, p. 21 If.; Pratt, 1964; Arrow, 1971) absolute risk aversion may be measured by

U(W)

ARA= Jt(^}

(6.16)

In the case of the second-degree utility function assumed in Equation (6.15), the ratio defined in Equation (6.16) would be

The direction of change of this ARA as wealth increases, or d(ARA)/dW, can be shown to be J(ARA)

4c2

>{)

(618)

The function therefore exhibits increasing absolute risk aversion. Similarly, relative risk aversion has been defined with reference to the proportion of a wealth portfolio that is invested in risky assets as the total level of wealth increases. This has been defined as (6.19) An investor who places a greater percentage of his wealth in risky investments as his total wealth increases is said to exhibit decreasing relative risk aversion. By following the same procedure as before, it can be shown that the illustrative wealth utility function in Equation (6.15) exhibits increasing relative risk aversion. By analogy, a second-degree, or quadratic, utility function such as we assumed in Equation (6.5), or the so-called Tobin function, exhibits the properties of increasing absolute and relative risk aversion. This, it should be noted, casts some doubt on the relevance in actual fact of the widespread use of this form of utility function in the literature on financial asset optimization. It would seem that contrary to the logical implications that we have just deduced, investors in general do in fact place larger amounts of their wealth in risky assets as their wealth increases, or, in other words, they exhibit decreasing absolute risk aversion. Less certainty attaches to the behavior in actual

6 Utility, uncertainty, and theory of choice 119 fact of relative risk aversion. But the quadratic utility functional form has been extensively used, and risky asset choice theory has been substantially based on the implicit mean-variance assumptions. Moreover, it has been shown that quadratic functional forms can be found that do exhibit more desirable risk-aversion properties (see Mossin, 1973). Axiomatic basis of stochastic utility The expected utility theory we have examined is derived from the work of von Neumann and Morgenstern, whose propositions regarding the axioms of choice in risky situations have influenced the vast literature that has developed in this field (von Neumann and Morgenstern, 1953).l In what follows, we shall specify the axioms by making direct reference to the selection of portfolios of risky assets. A particular portfolio, designated as Piy will be described by a set of possible rate-of-return outcomes, associated with each of which is the probability of its occurrence. In Equation (6.20), the /th portfolio available to the investor is envisaged as promising n possible rates of return, /?i through Rn, with associated probabilities px through pn. In the following discussion, the uppercase Pt refers consistently to a possible portfolio and the lowercase pt refers to a probability magnitude. Pi=[(Ri,Pi),

(R2,P2), . . . , (Rn, Pn)]

Pi^O, 2><= 1

(6.20)

To describe the ordering of choices among the various portfolios Ph we shall adopt the following standard notation: Pi>Pj indicates that portfolio Pt is preferred to portfolio Pj, and Pt~Pj means that the individual whose preference ordering is being considered is indifferent between portfolios Pz and Pj (see also Luce and Raiffa, 1957, Ch. 2; Hey, 1979). Axiom 1. Axiom of completeness or comparability Either Rt ^ Rj or Rj >: Rt for all /, j , = 1 , . . . , « . This axiom, which might be referred to also as the axiom of pairwise comparison, orders all possible pairs of the underlying rates of return on a portfolio. It states that the investor is able to describe a preference ordering over portfolio rates of return, such that his preference or indifference between all possible pairs of such rates of return can be established. Axiom 2. Axiom of transitivity of portfolio choices If Pi ^ Pj and Pj £ Pkt then Pt £ Pk. 1

The development given here follows that in Ford (1983, p. 18f.). A somewhat condensed treatment of the same subject is contained in Henderson and Quandt (1971, p. 42f.).

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II The neoclassical tradition

This axiom states, for example, that if the ith portfolio is preferred to the 7th portfolio, by virtue of its risk-return profile, and the7th portfolio is preferred to the Ath, then the Ufa portfolio will be preferred to the kth. These relations are assumed to hold for all pairs of portfolios in the set of portfolios available to the investor. Axiom 3. Axiom of continuity Assume that R\ is regarded as the most preferred rate-of-return outcome and that Rn is the least preferred. Then there exists a probability number, pit where 0 < / ? / < l , such that

Ri~l(Ri,pd, (Rn, 1 -pd]

for all 1= 1, . . . , n

This seemingly complex axiom will play a crucial part in the following construction of a utility function. It states that for each possible rate-of-return outcome Rt that may be designated (here shown on the left of the indifference sign), a combination of outcomes, and thus a portfolio choice, can be found that is indifferent to Rf. That portfolio would be made up of a combination of the most preferred and the least preferred outcomes, Rx and Rn, as shown on the right-hand side of the indifference sign. That is, there is some probability that could be assigned to the assumedly best possible outcome, with the magnitude of 1 minus that probability assigned to the worst possible outcome, that would make the individual indifferent between a combination of those best and worst outcomes and the designated outcome Rt. The magnitude of the probability pt that would have to be assigned for this purpose is dependent on the individual's attitude to risk and is therefore closely associated with the degree of risk aversion in his utility function. Axiom 4. Axiom of substitutability of outcomes and portfolios Let Rt and Qt be possible rates of return on a portfolio. If/?,-—Q,-, then Pi ~ P 2 , where l), Pl=

l(Rl,Pl),

( « 2 , / > 2 ) , • • • , (RhPi), (Rl,P2),

• • • , (Qi,Pi),

• • • > . . .

(Rn,Pn)] (Rn,Pn)]

This axiom makes a statement about the investor's attitude to two different portfolios, Pi and P2. The portfolios are understood to be the same in every respect except one. Portfolio Px contains a possible rate of return Rit whereas P2 contains a possible rate of return Qt. All other rates of return, and also their associated probabilities, are the same for both portfolios. If the investor is indifferent between the R; and the Qt rates of return, and if their probabilities of occurrence are the same for both portfolios, the investor will be indif-

6 Utility, uncertainty, and theory of choice

121

ferent between the two portfolios. This axiom is also referred to as the axiom of independence of irrelevant alternatives. Axiom 5. Axiom of monotonicity If, given Axiom 1, Rt ^ Rj and if

Pi = [(Rhpo)ARj, l-/?o)],and P2=[(Ri,Pi)ARj, l-/?i)],then Pi ^ P2

if and only if

/?o—Pi

This axiom states that an individual who compares two portfolios containing the same possible outcomes will choose that portfolio in which the preferred outcome has the higher probability of occurring. Axiom 6. Axiom of complexity IfPj=[(RhPij),

i = l , . . . ,n]

forj=l,

. . .,m

(that is, we suppose there exist m portfolios, Pp each of which contains n possible outcomes, Rh with associated probabilities/?/), and if (that is, there exists a "lottery" Si such that one of the portfolios Pj will be the prize in the lottery, with probability sj), and if (that is, there exists a portfolio S2 that contains n possible outcomes R[ with probability/?!), and if m

Pi= 2 ptjSj for i = 1, . . . , n, 7=1

then S\ This complex axiom is stating that probabilistic opportunities can be combined, depending on the probabilities specified, in such a way that an individual will be indifferent between (i) a given portfolio and (ii) a lottery in which one of the set of portfolios will be obtained as a prize. Interpretation of this rather complex case is assisted by focusing on the implicit probabilities of realizing the different possible portfolio rates of return by choosing (i) what we might call the direct route of S2 above or (ii) the indirect route of the lottery

Si. If the direct route of S2 is chosen, the probability of realizing a designated rate of return, Rh is, by definition above, pif which is equal, again by defini-

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II The neoclassical tradition

tion, to ^PijSj. Consider now the probability of realizing that same rate of return, Rh if the indirect route of S\ is chosen. In that case, we must consider the set of conditional probabilities involved. The first possible way of realizing R( is to obtain portfolio 1, Pu as the outcome of the lottery and then to realize Rt as the outcome of that portfolio. The probability of realizing Rt by this route, then, is the product of the probability of receiving Px from the lottery and the probability of Rt occurring conditional on the portfolio P\ having been obtained. A similar statement could be made regarding the probability of realizing /?, following the indirect result of each of the other possible lottery outcomes. The actual resulting probability of realizing /?, via the indirect lottery route will then be the sum of all of the products of probabilities thus obtained. We can write, therefore, piRf. S2) ~Pi— 2 Pusjas defining the probability J;= l of Rt via the direct route, and i = l , . . . ,n,j=l,

=p(Pi)p(Ri:Pi)+p(P2)p(Ri:P2)+ . . + Smpim

. . . ,m

•• •

jj

7=1

thereby defining the probability of/?/ via the indirect route. This, therefore, establishes the equality between the probabilities of portfolio outcome /?,- by either the direct or the indirect routes. The meaning of Axiom 6, on the basis of the assumptions specified, is accordingly established. Construction of the utility function In the following section we shall provide a detailed example of the construction of a utility function based on behavior in accordance with the foregoing axioms. An individual will be assumed to be acting rationally when making a choice between probability distributions, or between portfolios Pit if he acts in accordance with the axioms. This will imply that the individual will act in such a way as to maximize an ordinal utility function, where the utility of a portfolio can be written as U(P) and where that utility is a weighted sum of the utilities attached to each of the possible rate-of-return outcomes contained in the portfolio. These utilities of rates of return are weighted by the probability of their realization. We can therefore write U(P)= ^PiUi 1= 1

where ui=U(Rl).

(6.21)

6 Utility, uncertainty, and theory of choice

123

Let us lay the background for that discussion as follows. We now take the least preferred and the most preferred rate-of-return outcomes and assign to them arbitrary numbers on an ordinal utility scale over which it is desired to define the utility function. We may arbitrarily assign the utility value 0 to the least preferred outcome and the utility value 1 to the most preferred, designating these outcomes Rn and Rl9 respectively, as in Axiom 3. We now wish to find a way to allocate a number on the utility scale to any possible outcome between these least preferred and most preferred outcomes. Let such an outcome be designated Rt. Then we are assured by the axiom of continuity, Axiom 3, that it is possible to discover a probability magnitude such that Ri-KRupdARn,

I-Pd]

(6.22)

In accordance with the axioms, the individual is indifferent between the designated outcome Rif if it were available to him with certainty, and the chance of receiving either Rn or R\ with probabilities 1 —pt and ph respectively, as on the right-hand side of Equation (6.22). Both of these options must therefore be assigned the same number on the utility scale. Suppose that in relation to Rt the probability magnitude that established the indifference in Equation (6.22) was 0.75. Then we can substitute this probability magnitude into the right-hand side of Equation (6.22) and calculate the expected utility of the distribution of outcomes described there. That expected utility will be equal to the sum of (i) the utility that has been assigned to the most preferred outcome, namely 1, times the probability of realizing that outcome, namely 0.75, and (ii) the utility assigned to the least preferred outcome, namely 0, times the probability of realizing that outcome, namely 0.25. The expected utility calculates to 0.75, and that, therefore, is the utility magnitude to be assigned to the designated outcome of/?/. In a similar way, utility magnitudes could be calculated, relying repeatedly on the axiom of continuity, for assignment to all possible values of outcomes between the least and the most preferred. In this way, a utility function will have been defined over the entire range of possible outcomes. Once we have a utility function defined over the range of possible outcomes, it follows that the expected utility of a portfolio whose possible outcomes are described by a given probability distribution will be equal to the weighted sum of the utilities of each of the possible outcomes of the portfolio, where each such possible utility is weighted by the probability of its being realized. In other words, we have U(P)= I.piUiRd

(6.23)

i= 1

The investor, therefore, should select that portfolio that maximizes his expected utility in this sense.

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II The neoclassical tradition

The utility function, of course, must be consistent with the preference axioms. That is, we require that U(P i) > U(P2)

if and only if P { > P2

(6.24)

It can be shown from the axioms stated that this condition will be satisfied. Example of a derived utility function In the interest of concreteness, consider the following example of the derivation of a utility function defined over a range of possible outcomes. To change the area of application, let us imagine that a firm faces the prospect of a cash flow that may range between -$10,000 and $10,000. We desire to define the decision maker's risk-aversion utility function over that range. We use for this purpose the construction in Figure 6.5. On the horizontal axis of Figure 6.5 we have inscribed the range of possible outcomes, and on the right and left vertical axes, respectively, we have indicated measures of probability and utility. That is, we have arbitrarily assigned the point 0 on the utility scale to the utility of -$10,000 and the point 1 to the utility of $10,000. Imagine now that we desire to assign a point on the utility scale to the amount of $6,000. We proceed, in the manner of the previous discussion based on the axiom of continuity, to visualize a so-called reference lottery in which there are only two possible outcomes equal to the worst outcome of the possible cash flow being examined, or -$10,000, and the best possible outcome, $10,000.

-10

Figure 6.5

6 Utility, uncertainty, and theory of choice

125

We can deduce what would have to be the probability of success in the reference lottery, that is, the probability of receiving the $10,000 outcome, to make the expected value of the lottery equal to $6,000. This so-called breakeven probability follows from the equation p(10,000) + ( l - p ) (-10,000) = 6,000

(6.25)

The desired value of p is 0.80. Similarly, the probability of success in the reference lottery that would make the expected value of the lottery equal to, say, $4,000 is 0.70. If the probability of success in the reference lottery is 0.30, its expected value will be equal to —$4,000. In this way, we can build up a locus of "breakeven probabilities" that indicate the probabilities of success at which the expected value of the lottery will equal any designated amount within the range of possible outcomes of the cash flow, -$10,000 to $10,000. This locus of breakeven probabilities will coincide with the diagonal of Figure 6.5. We now suppose that the individual has received a certain cash flow of $6,000, and we ask him what would have to be the probability of success in the reference lottery to make him indifferent between retaining the $6,000 and surrendering it for a ticket in the lottery. He knows, of course, from the foregoing that the expected value of the lottery will be precisely equal to his $6,000 if the probability of success is 0.80. If, however, he exchanged his $6,000 for the lottery ticket, he would stand to gain a further $4,000 but would stand to lose $16,000, equal to the $6,000 he surrendered for the lottery ticket plus the $10,000 he would have to pay if the "lose" rather than the "win" result of the lottery occurred. Being a risk averter, as we shall suppose, the individual would state that he would require a probability of success greater than the breakeven probability to make him indifferent between his $6,000 and the lottery. Let us suppose that after a certain amount of questioning, meditation, and answering, the individual stated that the indifference would be established, and he would be prepared to exchange his $6,000 for the lottery ticket, if the probability of success were 0.90. In the same way, we might establish that the individual would be indifferent between the sum of $4,000 and the lottery, not if the probability of success were simply the breakeven probability of 0.70 but were, say, 0.85. If the individual had received zero dollars and he were invited to exchange it for the lottery ticket, or, that is, to accept a free ticket in the lottery, we can similarly ascertain the probability of success that would make the exchange possible. If, of course, the individual did accept the lottery ticket, he would stand to gain $10,000 but would also stand to lose $10,000. His risk aversion would, presumably, cause him to require a probability of success greater than the breakeven probability of 0.50. Let the required probability be 0.70. Finally, as an example on the other side of the range of possible outcomes of the cash

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II The neoclassical tradition

flow, we might imagine that an amount of -$4,000 had been received, or that a loss or a debt of $4,000 had been incurred. If the individual were to exchange that amount for a lottery ticket, the lottery organizer would, in exchange, be agreeing to relieve him of his debt of $4,000. If he were successful in the lottery, his net gain would be $14,000, equal to the sum of the $4,000 debt of which he had been relieved and the $10,000 prize in the lottery. But if the "loss" outcome of the lottery occurred, he would realize a net loss of $6,000, equal to the $10,000 he was now required to pay to the lottery less the $4,000 debt of which he had been relieved. His risk-aversion characteristics again will probably make him require a probability of success greater than the breakeven probability of 0.30. We can suppose it is 0.50. In Figure 6.5, these required probabilities of success in the reference lottery are shown above the relevant magnitudes on the horizontal axis. The same procedure may be followed with relation to any other magnitude in the cash flow range between -$10,000 and $10,000. If, then, these probabilities are joined as in the figure, we have what we shall call a locus of "indifference probabilities." It will be clear that the concavity of the indifference probability locus is due to the degree of risk aversion in the mind of the individual whose utility function is being derived. Moreover, the greater the degree of risk aversion, the greater will be the concavity in the indifference locus. If the individual were risk indifferent, he would have been satisfied at every point of the foregoing to exchange given amounts of cash for the lottery ticket so long as the breakeven probabilities were operative. The locus of indifference probabilities would then be linear and would coincide with the locus of breakeven probabilities. Finally, if the individual were a risk lover rather than a risk averter, his indifference probability locus would be strictly convex, or would swing down along the southeast of the breakeven probability locus in Figure 6.5. We are now ready for the final step that transforms the indifference probability locus into the desired utility function. Suppose, for example, that we wish to assign a utility number to the cash flow outcome of $6,000. We have just seen that the individual is indifferent between $6,000 and the lottery with a probability of success of 0.90. This indifference, then, requires us to assign to the amount of $6,000 the same utility number as represents the expected utility of the lottery with a probability of success of 0.90. That may be calculated as follows: E[U(L)] = 0.9(H/($10,000) + 0. \0U($ - 10,000) = 0.90(1)4-0.10(0) = 0.90

(6.26)

The utility number to be assigned to the amount of $6,000 is therefore 0.90. In a similar way, utility numbers may be derived for any amount within the

6 Utility, uncertainty, and theory of choice

127

range of cash flows between -$10,000 and $10,000. By assigning the arbitrary numbers of 0 and 1 to the limits of the range of possible cash flows, we have been able to read the indifference probability locus against the right vertical axis, or the probability axis, of Figure 6.5 and to read the same locus as the derived utility function against the left vertical axis, or the utility axis. This convenient result, of course, is due only to the arbitrary assignment of the utility range that we made. Any other numbers could have been assigned, as the utility function we have derived is, as we have indicated, an ordinal function. If the assumptions contained in the axioms we have defined are satisfied, and if individual behavior reflects the statement of the axioms or is in that sense rational behavior, the axioms permit us to conceive of a utility function of the general form shown in Equation (6.5). There, as in the instance we have just examined, the degree of concavity reflects the degree of risk aversion in the mind of the decision maker. Uniqueness of the utility function A significant difference exists between the ordinal utility function in terms of which we discussed consumer behavior earlier in this chapter and the utility function we have now derived. The former, we have seen, is unique up to a monotonically increasing transformation. But in the present analysis it is possible to envisage monotonically increasing transformations that, if they were applied to the investor's utility function, would not preserve a consistent ranking of portfolios. As a simple case, consider the following example, based on the information in Table 6.1. Column 2 of the table shows the utilities assigned to the range of values indicated in the first column, thereby describing an arbitrarily assigned ordinal utility function U\. In column 3 are shown the utility magnitudes that would apply if a monotonically increasing transformation of the first utility function were adopted, that is, U2 = (Ui)2. Let us suppose that two investment opportunities exist: Si, which offers a Table 6.1 Ordinal utility functions Dollars

Ux

U2 = (U{)2

0 2 4 6 8 10

10 40 65 85 100 110

100 1,600 4,225 7,225 10,000 12,100

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II The neoclassical tradition

50 : 50 chance of receiving $2 and $10, and S2, which offers a 60 percent chance of receiving $4 and a 40 percent chance of receiving $8. From the second column of Table 6.1, describing U\, the expected utilities of S\ and S 2 are = 0.50(40)+ 0.50(110) = 75

(6.27)

E[U(S2)] = 0.60t/($4) + O.4 = 0.60(65)+ 0.40(100) = 79

(6.28)

It follows that 52 will be preferred to Si on the basis of its higher utility ranking. If the two investment opportunities are evaluated against utility function U2, the utility assignments are 6,850 for Si and 6,535 for S2. The order or ranking will now be reversed, and Si will be preferred to S2. Thus we see that a monotonically increasing transformation of the original utility function over the range of monetary values specified in the example may fail to be order preserving. It can be shown, however, that a monotonic linear transformation of the ordinal utility function will preserve the preference ordering in several important respects. Let us suppose a utility function of the form that provides, in accordance with Axiom 3, UB =PiUA + (1 ~Pi)Uc for some pt. Recalling that Y is a monotonic linear transformation of X if Y=a + bX, with ft>0, we can transform the utility function U to the function (6.29) It follows from this that U = (U*- a)lb, orU = cU* + d, where c = I/ft and d= —alb. We can say, therefore, that U B in the present example transforms into cU%-\-d. Following the example, we can therefore write +d

(6.30)

from which it follows that cU%=PicUX + c{\-pdUt

(6.31)

and therefore U$=p,UX + (l-pi)Ut (6.32) This demonstrates that the monotonic linear transformation of the original utility function given in Equation (6.29) provides the same results as the original utility function and is therefore itself a utility function.

6 Utility, uncertainty, and theory of choice

129

The utility functions provided by the von Neumann-Morgenstern analysis that we have followed up to this point, although they are in a basic sense ordinal, have to be regarded as cardinal in a restricted sense. They partake of the same limitations of ordinal measures that the consumer utility functions contain. They do not permit interpersonal comparisons of utility or permit the statement that an outcome providing a utility of, say, 70 is preferred twice as much as an outcome with a utility measure of 35. This follows, of course, because the choice of the origin of the utility scale is arbitrary. The von Neumann-Morgenstern utility numbers do, however, provide an interval scale such that the differences between the utility numbers are meaningful. This follows from the fact that a monotonic linear transformation preserves the relative magnitudes of differences between the numbers on the original utility scale. That is, if the difference Uc - UB is greater than UB — UA on an original scale, the same relative magnitudes of differences will be preserved under the monotonic linear transformation. Under a monotonic linear transformation, also, the sign of the rate of change of marginal utility, or the second derivative of the utility function, is invariant. The von Neumann-Morgenstern ''ordinal-cardinal" utility function, at the same time as it permits this comparison of utility differences, also permits the calculation of expected utilities in the sense in which we explored that earlier in this chapter. It thus leads to the decision criterion of expected utility, and it facilitates decision making under risk or the discrimination between stochastic objects of choice. The firm's choice of output under selling price uncertainty An application of the expected-value concept discussed in this chapter is found in the firm's choice of its optimum output when uncertainty attaches to the price at which that output can be sold.2 A brief digression on this problem will consolidate our understanding of many of the main points at issue in the following chapter. We assume that in a simple static model the firm is required to choose its output level x before the selling price p is known but on the understanding that the price will be determined in a perfectly competitive market. The firm's total cost function is assumed to be known as TC(JC)=A + C(JC)

(6.33)

where A represents the fixed costs and C(x) is the variable-cost function. The firm's decision maker possesses a utility function defined over profits, TT, where 7r=px-A-C(x) 2

(6.34)

The argument in the text is based on Sandmo (1971) as reported in Ford (1983, p. 173f.).

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II The neoclassical tradition

As in the preceding sections of this chapter, the probability distribution of the selling price is assumed to be assigned by the decision maker and is denoted We can then make use of the expected utility therorem to specify the firm's expected utility of profit as E[U(7T)] = ^U[px-A-C(x)]f(p)dp

(6.35)

where p is assumed to be nonnegative, as it represents the market price of the output commodity. By differentiating Equation (6.35) with respect to x, we find the first-order condition for a maximum value of E(U). This is accomplished by first differentiating the utility term under the integral sign, applying the chain rule in doing so, to yield

F

{

\

dp

(6.36)

Equation (6.36) will be recognized as describing the expected value of the term under its integral sign to the left of the probability function [analogous to Equation (5.4) of Chapter 5]. We can therefore write, as the first-order maximization condition,

E[u'(7T){p-C'(x)}] = 0

C(X)>0

(6.37)

If, further, we differentiated Equation (6.36) with respect to x, we should have the second-order condition for the utility maximum: E[U"(TT){P-C'(X))2-

t/'(7r)C"(Jc)]<0

(6.38)

In the second-order condition in (6.38), the signs above the elements of the expected value indicate the signs they can be assumed to take. The sign on U"{TT) will be negative in the case of a risk-aversion utility function, as developed at length earlier in this chapter. The sign on the squared term is, of course, positive. That on U'(TT) will be positive under the axiom of monotonicity. But it follows that the sign on C"(x) will not necessarily have to be positive in order to establish the negativity condition of the overall expression in (6.38). Under certainty, of course, the positivity of the second derivative of the cost function would have been necessary, as is established in the usual textbook case. A principal conclusion can now be derived from this model. In the case of uncertainty, it can be shown that E(p) § C'(x)

according as U"(if) § 0

(6.39)

6 Utility, uncertainty, and theory of choice

131

That is, the output volume will be such, if the decision maker is a risk averter, that the expected price is greater than the marginal cost. We can explore this condition in the following way. First, we take the expected value specified in Equation (6.37). Here we have the expectation of a product term. It will be helpful, therefore, to return to the basic discussion of expected values and covariances in Chapter 5 and observe the following. Let us assume that X and Y are random variables and that we desire to specify the expected value of their product, or E(XY). Recalling a similar development in Chapter 5, we can write Xi = E(X) + dx

and Yt = E(Y) + dy

Then

E(XY) = E[[E(X) + dx){E(Y) + dy}] = E[E(X)E(Y) + E(X)dy + E(Y)dx + djy\ Placing the first expectational operator within the square brackets and recalling that E(dx) and E(dy) both equal zero and that Eidxdy) provides the covariance between X and Y} it follows that E(XY) = E{X)E(Y) + Cov{XY)

(6.40)

We can then apply this procedure to specify the expected value indicated in Equation (6.37) as required above. It follows that E[U\TT){P

- C'(jt))] = E[U'(ir)][E(p) - C(x)] + Cov[t/'(7r),p]-Cov[£/'(7T), C(x)]

(6.41)

In order to justify the conclusion stated in (6.39), let us note that if, say, U"(TT) as stated in that conclusion is negative, the covariance between U'(TT) and p in Equation (6.41) will be negative and that between U'{TT) and C'(x) will be positive. These results hold because if, for example, the price of output increases, thereby increasing the profit level under the assumed conditions of perfect competition, the marginal utility of profit will diminish because of the negativity of the second derivative of the utility function. Therefore, the covariance between the price level and the marginal utility of profit will be negative as stated. Similarly, if, for any given price level, the level of marginal cost should increase, this will diminish the profit level and thus imply an increase in the marginal utility of profit. Thus the covariance between the marginal cost and the marginal utility of profit will be positive as stated. These conditions, taken together, imply that when the marginal utility of profit is diminishing, or U"(TT)<0 in (6.39), or the firm's decision maker is a risk averter, the last two terms on the right-hand side of Equation (6.41) will be negative. In order to establish the maximum condition in Equation (6.37), therefore, the first term on the right-hand side of Equation (6.41) must be

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II The neoclassical tradition

positive. That implies, however, that E(p) must be greater than C'(x). This, then, establishes the condition specified in (6.39) as the results of the price uncertainty case, as opposed to that of the general textbook case of perfect competition under price certainty. This means that if E(p) should equal the fixed price that would have ruled in the market under conditions of certainty, the output will be lower under uncertainty [in order to make E(p)>C'(x)] if the firm's decision maker is a risk averter, that is, if U"(TT)<0. If the decision maker is risk netural, the outputs would be the same under both certainty and uncertainty, and if he is a risk lover, with U'XTT) positive, the output will tend to be higher under uncertainty.

CHAPTER 7

Financial asset markets and the cost of money capital The problem of money capital, we have argued, needs to be integrated into the theory of the firm at as early a stage as possible. To employ the categories that the neoclassical theory inherited from Marshall, it needs to be incorporated before the beginning of the short-run, rather than after the end of the long-run, analysis. Otherwise, money capital has no analytically integrative significance for the theory of the firm at all. Moreover, the notion of money capital, together with that of the money capital availability constraint, imports into the theory of the firm the realities of uncertainty and the lack of perfect expectation and foresight. The cost of money capital is very much an uncertainty concept. It takes up the questions of intertemporal valuation and the specification of risk-adjusted discount rates, or the rates of return required by the risk-averting suppliers of money capital at which future income streams should be reduced to their economic values. Behind the neoclassical development of these concepts lies a highly developed theory of the money capital or financial asset market. The cost of capital depends, in that body of analysis, on the prices of risky assets and the rates of return they provide when equilibrium conditions in the asset market are satisfied. The apparatus we need in order to consider this equilibrium theory of prices and yields has been almost completely assembled in the preceding chapters. The theory is heavily indebted to the thought forms of the probability calculus. It interprets the rates of return on the securities that represent claims to the firm's income stream as random variables describable by subjectively assigned probability distributions. In order to examine the theory and its implications, we set aside for the present the doubts we have raised as to whether the probability calculus is as applicable in economics as it is in other fields of enquiry. In this chapter we shall examine the neoclassical development and shall return to alternative constructions in Part III. Assumption content of the equilibrium theory of financial asset prices The theory of financial market equilibration is a development of the two-asset portfolio theory noted in Chapter 5. In expanding this to consider the general n-asset case, we shall develop a static theory of the equilibrium market prices 133

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II The neoclassical tradition

of a given stock of financial assets. These may be imagined, for example, as the common stock securities of nonfinancial corporations, such as those listed on the New York Stock Exchange. The existing theory has enjoyed considerable development beyond the essentials we shall examine in what follows. But it is not necessary or desirable for our present purposes to attempt a full exposition of the nuances and extensions that have entered the literature.1 The theory is seen as a "general equilibrium model of the pricing and allocation of the given supply of securities . . . (which) describe(s) the characteristics of a situation in which investors' combined demand is equal to the supply for every security" (Mossin, 1973, p. 64). The analysis proceeds in the tradition of general equilibrium theory. In fact, Fama and Miller, two of the most articulate and influential architects of the theory, have observed that "we use the term 'market equilibrium' in recognition of the fact that we always work conditional on an assumed equilibrium in the markets for labor and consumption goods. . . . In essence, we examine the characteristics of capital market equilibrium, given a general equilibrium, that is, simultaneous equilibrium in all markets" (1972, p. 279n.). In the application of the theory to the financial decisions of the firm, it is generally assumed that "the market is always near equilibrium" and that "the market will move toward equilibrium if a disequilibrium condition sets in" (Bierman and Hass, 1973, p. 230). The asset market theory in its standard Sharpe-Lintner-Mossin form is based on the following further assumptions. 1.

2. 3.

1

All investors in the financial asset market are single-period (all having the same single-period time horizon) expected utility of end-ofperiod wealth maximizers who choose among alternative portfolios of asset opportunities on the basis of the mean (expected value) and variance of the rate of return. All individuals are optimizing over risk-aversion utility functions. All investors can borrow or lend an unlimited amount of funds at an exogenously given risk-free rate of interest, RF. All investors have made identical estimates of the means, variances, and covariances of returns among all available assets.

See the references in Chapter 5 to the work on the foundations of the theory by Markowitz, Sharpe, Lintner, Arrow, Tobin, and Hicks. Important development and application is contained in Fama and Miller (1972) and Mossin (1973). More recent developments are contained in Weiss (1984) and his rejoinder to Vickers (1984). A highly valuable discussion of what he terms "the orthodox approach" is contained in Ford (1983, Ch. 1). See also a number of more advanced treatments of related issues in the papers included in Levy and Sarnat (1977). Note there the paper by Arditti and Levy, which extends the analysis beyond the two-moment, or the mean-variance, case considered in this chapter. Further expansion of the theory is contained in Elton and Gruber (1984), and applications to the firm's cost of capital and capital expenditure decisions are provided in Brealey and Myers (1984).

7 Asset markets and cost of money capital 4. 5. 6. 7.

8. 9.

135

All assets are perfectly divisible and are marketable free of transactions costs. There are no taxes. All investors are price takers in a perfectly competitive asset market. The total quantity of all assets in the opportunity set is given and fixed, each such asset being uniquely defined, as implied by assumption 3, by a universally agreed, subjectively assigned probability distribution of returns. Each individual enters the market with a given wealth endowment, which he proceeds to distribute over the asset opportunity set in a manner that is optimal in the utility-theoretic sense. No changes occur in the asset opportunity set during the single-market period, it being assumed that corporate firms neither issue new securities nor withdraw previously outstanding securities from the market.

Only brief comments need be made on the assumptions at this point. The first assumption recalls the analysis of probability distributions and utility theory in Chapters 5 and 6. The second assumption is analogous to that included in the development of the separation theorem in Chapter 3, where it was assumed that both firms and the owners of firms could borrow or lend at a given and exogenous market rate of interest. The separation theorem in what follows depends also on the third assumption, that of "homogeneous expectations." This, it will become clear, is necessary in order to obtain from the theory a uniquely definable, "utility-free," optimum portfolio of risky assets. Attempts have been made to dispense with the assumption of homogeneous expectations, for example by Lintner (1969; see also Jensen, 1972, p. 390), but in such cases theoretical complications of a high order follow. The assumption of perfect financial asset markets is expressed in both the perfect divisibility and marketability of assets and in the unlimited borrowing and lending opportunities at a risk-free rate of interest. Difficulties inhere in the assumption of given and uniform borrowing and lending rates, as they do also in connection with the givenness of the asset opportunity set. In the movement and change that continually occurs in the economy, firms also change. They expand, merge, decline, and die, and the volume and kinds of their capital securities outstanding also change, with traceable implications for financial investors' market opportunities. Similar observations might be made regarding the assumption of the givenness of investors' wealth endowments, and the notion that transactions always take place in the asset market at equilibrium prices. The possibility of "false trading," or the consummation of market transactions at nonequili-

136

II The neoclassical tradition

brium prices, is assumed away (see Hicks, 1946, p. 127; Vickers, 1975, 1978). If false trading did occur, redistributions of endowments would take place prior to the achievement of market equilibrium, and this could conceivably have further expectations-inducing effects. But at this stage we are interested in the validity of the conclusions that follow from the assumptions of the theory. The determination of equilibrium asset prices will provide the equilibrium yields, or the market's required rates of return, on those assets. It is in this form that the theory leads to a description of the firm's cost of money capital. Portfolio asset choice The economic value of the ownership or equity capital in the firm is defined by the present discounted value of the firm's expected time vector of residual earnings. Using notation employed up to this point, the value of the firm can be written as V— irlp. We now need to develop an explanation of what it is that determines the discount rate or the capitalization rate, p, in this valuation formula. Generalizing from the two-asset portfolio case in Chapter 5 and making use of the relations between expected values, variances, and covariances between rates of return on financial assets, we define the expected rate of return and risk on an n-asset portfolio. In the following development, we shall use the symbols Rt and Rj to refer to the rates of return on the /th and the jth assets or securities in a portfolio, and Rp will refer to the portfolio rate of return. The weights attached to the portfolio assets, here referred to as xit xjt will represent the proportion of the total portfolio value accounted for by the /th and y'th assets. The portfolio weights will therefore sum to unity. The expected value, variance, standard deviation, and covariance notation will be recognized from preceding arguments, and the coefficient of correlation between pairs of asset returns will again be denoted by p,y. For the «-asset portfolio, E(RP)= $XiE(Rd

2^=1


£ XiXjPijCTiaj

(7.1) (7.2)

i=\j>i

or aiRp) = [lSaiXjaij\^

(7.3)

It is possible to generate, in the case of this Az-asset opportunity set, an efficient boundary that is of the same nonlinear kind as the two-asset opportunity locus derived in Chapter 5 under the assumption that the underlying

7 Asset markets and cost of money capital

137

E(R)

a(R)

Figure 7.1 asset returns were less than perfectly correlated. We saw then that the form of such an opportunity locus depended on the form of the covariance matrix describing the relations between the asset returns. That is, it depended on the magnitude and sign of the covariances, or the coefficients of correlation, between all possible pairs of assets that were candidates for inclusion in the portfolio. The n-asset opportunity set therefore generates the nonlinear boundary of the form depicted by the curve EE' in Figure 7.1. The locus EE' in Figure 7.1 is referred to as the efficient portfolio opportunity boundary. Every point on that locus dominates attainable portfolios interior to the opportunity set. The risk-return plane is populated by a set of points, each of which describes the risk-return coordinates of a separate asset opportunity or an achievable portfolio combination of assets. Point M, for example, is one such point and indicates that an asset is available that provides an expected return and standard deviation of return described by the intercepts to the axes from that point. Every portfolio on the boundary between R and S dominates portfolios in the interior of the segment RMS, in the sense that it offers a higher expected rate of return for a specified degree of risk, or a lower degree of risk for a given expected return. In this sense, the boundary EE' is itself undominated. The efficient locus EE' is objectively calculable in the manner we shall indicate immediately below once we are given the forms of the subjectively assigned probability distributions of the underlying asset returns and the relevant covariance matrix. The locus EE' can then be regarded as the objectively attainable asset market risk-return trade-off. But it is, of course, objec-

138

II The neoclassical tradition

tively attainable only in the sense that the actions of market participants consistent with their assumed probability beliefs about asset returns will clear the asset market in the manner we shall investigate. If those beliefs should be falsified, market movements of a new and different kind will point to a new and differently specified market clearing condition. The slope of the EE' locus at any point, dE(R)/dcr(R), describes the objectively attainable marginal rate of transformation between risk and return. Any point on the locus is accordingly a potential equilibrium point for the investor. We solve the problem of asset choice by setting out to minimize Equation (7.3) subject to Equation (7.1). We specify a required rate of return and then determine the portfolio combination of assets that would provide the minimum attainable risk, or standard deviation of return, at the same time as it offers the specified rate of return. To discover this portfolio, it is sufficient to solve for the magnitudes of the xt weights that would provide the desired asset combination. We visualize the constrained minimization problem in the form mm + \ 2 [ 1 - 2>/]

xt ^ 0 for all /

(7.4)

The first constraint on the right-hand side of Equation (7.4) anchors the solution to a specified rate of return, and the second constraint takes account of the fact that the weights must sum to unity. If a portfolio rate of return, E(RP), is specified in the first constraint and the solution values of the xt are calculated, the resulting portfolio will be described by a point on the efficient portfolio locus such as EE' of Figure 7.1. By varying the level of E(RP) to which this first constraint is anchored, the entire portfolio locus can be generated (see Sharpe, 1963, 1970, for computational procedures, and Elton and Gruber, 1984, Chs. 5-7). The meaning to be attached to the risk of an asset needs further clarification in this context. An asset's own, or intrinsic, risk has been defined in terms of the variance of the subjectively assigned probability distribution of that asset's rate of return. But when we take account of the contribution that that asset makes to the portfolio's covariance matrix, we must envisage also what we shall term the asset's portfolio membership risk. It is this latter concept that we shall have in mind when the risk of an asset is referred to in the following analysis. The risk of an asset is measurable, then, by the contribution a unit of that asset makes to the risk of the portfolio of which it is a member, or by There will be occasion to use in what follows two other equivalent expressions for the same idea. Thus,

7 Asset markets and cost of money capital

139

_(r2(RP)


from which it follows that dXi

CT(RP)

From a different viewpoint, the covariance between an asset's return and the return on the portfolio can be written as 2jXjRj) = 2jXfCo\(Rh

Cov(/? ; , Rp) = Cov(Rh

/?.•)

Substituting this into the numerator of Equation (7.5) provides da(Rp) _Cov(Rh Rp) dXi

a(Rp)

(7.6)

We then have three formulations of the risk of an asset: (7.7)

\lxp)

u

Co\(Rt, R.) cr(RP)

(7.9)

Risk-free asset opportunity Consider again the concave portfolio opportunity locus. If the investor was required to invest his entire wealth endowment in the opportunity set of risky assets, he would be confronted with the task of deciding which of all the efficient portfolios he should choose from the points on the efficient boundary. But in fact, the investor may not be constrained to invest only in risky assets. It is assumed by the theory that he may place a part or the whole of his wealth endowment in a risk-free asset yielding a riskless rate of return RF. As shown in Figure 7.2, let us imagine that an investor was to divide his

140

II The neoclassical tradition E(Rp

o(Rp)

Figure 7.2 investable endowment between the risk-free asset yielding a return of RF and portfolio A, which promises a return of E(RA), not shown in the figure, and presents a degree of risk
(\-w)2a2(RA)

(7.11)

or (7.12) Differentiating Equations (7.10) and (7.12) with respect to w and dividing the first such result by the second yields dE(Rp) da(Rp)

E(RA)-RF a(RA)

(7.13)

In adopting this procedure, we have specified (i) the rate at which the portfolio expected return would be changed, in this case reduced, by an increase in w, or in the relative importance of the risk-free asset [by taking the derivative of Equation (7.10) with respect to w], and (ii) the rate at which the portfolio risk would be affected by the same iction [by taking the derivative

7 Asset markets and cost of money capital

141

of Equation (7.12) with respect to w]. The implicit relation between these changes in the portfolio expected return and risk, consequent upon the change in w, is described by Equation (7.13). It represents the manner in which the investor may take advantage of the risk-return trade-off that is available in the asset market. If the relative importance of the risk-free asset in the portfolio is increased in the manner we have just envisaged, this would imply a reduction in the portfolio risk and also, as the development leading to Equation (7.13) indicates, a reduction in the portfolio expected rate of return. Alternatively, we may solve Equation (7.12) for w to obtain

and substitute this into Equation (7.10) to obtain (7-14) Differentiation of this equation with respect to a{Rp) provides the result shown in Equation (7.13). Further, rearrangement of Equation (7.14) yields the result shown in Equation (7.15) that defines the slope of the attainable locus RFA:

E(RP)=RF + \E(R*i

RF

]a(Rp)

(7.15)

Given the risk-return characteristics of portfolio A, this provides a linear opportunity locus. The slope coefficient of that locus, defined by the bracketed term in Equation (7.15), is analogous to what will emerge below as the market price of risk implicit in the asset market portfolio opportunities. The existence of this opportunity implies that the new portfolio locus RFA dominates that portion of the previously existing locus EE' that lies to the southwest of A. In exchange for a risk exposure of 05, for example, the EE' locus offers an expected return of ST, whereas the new linear locus offers a prospective return at that risk level of SV, indicating a dominant portfolio. The introduction of the risk-free asset opportunity has therefore transformed the efficient portfolio boundary from EE' to an effective efficient opportunity locus of RFAE'. If, in the same way, the investor were to contemplate combinations of the risk-free asset and portfolio opportunity B, he could achieve a dominating locus of RFB and an effective opportunity locus of RFBE'. By experimenting in a similar fashion with other possible risky asset portfolios, the investor could discover an optimum portfolio of risky assets with which the risk-free asset could be combined. Such a portfolio is designated as M in Figure 7.2. At that point, the linearized locus is tangential to the

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II The neoclassical tradition

previously existing EE' boundary. The investor's effective opportunities are then described by RFME'. Such an attainable locus is dependent, of course, on the specified risk-free rate, RF. A different risk-free rate would generate a different opportunity locus and a different optimum portfolio of risky assets. But we have not at this stage considered what, in a general equilibrium asset market context, will determine that risk-free rate (see Vickers, 1975). Considerations of general economic conditions and monetary policy are clearly relevant. The risk-free rate will be dependent also on the volume of borrowing and lending that market participants desire to undertake at that rate. To the extent that the investor is allocating his wealth between the risk-free asset and the optimum portfolio of risky assets M, he is lending at the riskfree rate of interest. Depending on his decision as to the proportion of his wealth that he should lend at that rate, he will settle at a point on the linear segment RFM. But suppose also that the investor could borrow at the same risk-free rate. Then he could increase his expected return even further, at the same time undertaking additional risk, by borrowing at the risk-free rate and investing the proceeds in the risky portfolio M. In such an event, he would settle somewhere on the MZ portion of what is now a completely linearized opportunity locus RFMZ. This locus may now be referred to as the Capital Market Line. It should be recalled that in this analysis we are dealing with an equilibrium theory of financial asset prices and yields. Every investor in the market is contemplating the selection of an optimum portfolio from the given asset opportunity set and will similarly be concerned to discover the optimum combination of the risk-free asset, promising the given return of RF, and the optimum risky asset portfolio M. What are being discovered in such relations as those summarized in Figure 7.2, then, are the conditions that will exist when full general equilibrium conditions are satisfied in the asset market. This requires us to make a careful interpretation of the linear portfolio opportunity locus we have derived, such as RFMZ. It is, in a sense, objectively attainable, in the manner in which we previously referred to the efficient set boundary EE' as objectively attainable. But in both cases, that objectivity is based on the existence of underlying investor preferences that determine their demands for securities in the asset market and influence the determination and expectations of market prices and yields. Moreover, it would be a mistake to interpret the locus RFMZ as a trading opportunity locus in a disequilibrium opportunity sense, along which investors could move once it had been established. Rather, that locus describes simply the conditions that exist and the relations between risks and rates of return that exist when equilibrium conditions are satisfied. The theory of equilibrium asset prices, in other words, is not competent, in the form in which it has been developed, to make any

7 Asset markets and cost of money capital

143

statement about trading processes. It is a theory, not of trading behavior, but of equilibrium relations. We have assumed, in order to exhibit the results of this financial market theory, that all participants in the market hold the same expectations regarding the risk-return profiles of the asset opportunities. Every investor who holds risky assets will in that case choose to invest part of his wealth in portfolio My and we may then refer to this as the market portfolio. This portfolio's expected return and risk will be referred to in what follows as E(RM) and CT(RM), respectively. This risk and return profile of the market portfolio will be the outcome of the supply and demand forces of asset market equilibration that establishes the final asset prices. These, in turn, establish the asset yield relations that define the final opportunity locus. The slope of that locus, which defines the market risk-return trade-off that exists when full equilibrium conditions are satisfied', will be referred to as the market price of risk. The market portfolio M must be made up of all the assets in the risky opportunity set, and the proportion each asset occupies in the portfolio must equal the ratio of its market value to the total market value of all the assets combined. It is impossible for an asset, say the 7th asset in the set, to account for a proportion of the total market value of all risky assets that is different from the proportion of the value of their risky investments that all investors combined place in that asset. In full equilibrium, all the assets will be held by willing holders, the amounts borrowed will be offset by amounts lent, and net borrowing will be zero (see Sharpe, 1970, p. 82).

Asset market equilibrium condition Consider now the portfolio selection or the optimum-asset choice problem confronted by a representative investor and summarized in Equation (7.4). Differentiating this objective function with respect to xt and Xj and setting the partials equal to zero for first-order optimization conditions yields d_da(Rp) -\lE(Ri)-\2

a* * W

=0

, ^,_Xa=0

(7.16) (717)

It follows from Equations (7.16) and (7.17) that (7.18)

144

II The neoclassical tradition

or that (7.19) From the mathematical interpretation of Equation (7.4), Xi describes the partial derivative of the minimand function with respect to the constraint variable to which the Xi is attached, or Ki = dcr(Rp)/dE(Rp). Understanding, as summarized in the preceding sections, that every investor will choose to invest part of his wealth in the market portfolio when the market equilibrium conditions are satisfied, we can now investigate the nature of the portfolio optimization conditions at the optimum risky portfolio point M. In the further development of the conditions in Equations (7.16)-(7.19), and in the interpretation of the equilibrium solution value of Xi, we can replace E(RP) and o{Rp)byE(RM)2Lnd(T(RM). The solution value of Xi at the optimization point can be linked to the fact that doiRM)ldE{RM) is the reciprocal of the slope of the Capital Market Line, RFM, in Figure 7.2. This latter magnitude, MN/RFN in the figure, can then be interpreted as the market price of risk. It is equal to [E(RM — RF]loiRM), and it indicates the increment of return the market is providing in equilibrium for a unit change in portfolio risk. Focusing now on the relations that exist when the market portfolio is held, we may substitute this equivalent expression for X2 in Equation (7.19) and perform the following operation. First, multiply the resulting equation by xx and record the result. Then multiply the equation by x2 and record the result. Proceed in a similar fashion, multiplying the equation by each of the *,-. We then have a total of n such new equations. If all of these n equations are added, the following result is obtained:

^XiE(Rj) - ^XiE(Ri) = i

i

2*, — o-{RM)

L/

dXj

2*,- — i

(7.20) 0Xi

J

We recall now that 2;t/ in the first term on the left-hand side of Equation (7.20) equals unity, so that the term as a whole equals E(Rj). In the second term on the left-hand side, each JC, is multiplied by its corresponding E(Ri), as the summation takes place over the index /, and the term as a whole therefore equals the expected return on the market portfolio, E(RM). Similar interpretations attach to the 2;c; on the right-hand side of Equation (7.20). Now in making a further interpretation of Equation (7.20), we may substitute for the first term inside the brackets on the right-hand side the equivalent expression in Equation (7.9). For the final term inside the brackets we may substitute the equivalent expression in Equation (7.8). Making the substitutions, we obtain

145

7 Asset markets and cost of money capital

, RM )/Var {RM )

Figure 7.3

F(R,

F(R

,_E(RM)-RF\COV(RJ,RM) (T(RM)

4

(7.21)

The last term inside the brackets in Equation (7.21) can be recognized as the variance of return on the market portfolio divided by its own standard deviation, or simply the standard deviation of the market portfolio return. This equivalence permits us to remove the brackets from Equation (7.21), with the following result: E(Rj)-E(RM)=RF

-

rE(RM)-RF~\Cov(Rj,RM)

(7.22)

Equation (7.22) may then be simplified to provide E(RJ)=RF+IE(RM)-RF]COV(RJ'RM)

(7.23)

To make an interpretation of the final term on the right-hand side of Equation (7.23), and focusing on the expression Co\(Rj, R^I^RM), consider the relationship shown in Figure 7.3. Here we have envisaged the regression of the rate of return on the jth asset on the rate of return earned on the market portfolio. The regression coefficient in such a relationship may be written as the covariance between the two variables divided by the variance of the independent variable. The regression coefficient is then defined precisely by the

146

II The neoclassical tradition

M

1 1

RF

0

•

1

0.5

10

1.5

1

2.0

Figure 7.4 ratio we now observe on the extreme right-hand side of Equation (7.23). This magnitude has attracted a precise designation in the theory of capital asset market equilibrium and is referred to as the 7th asset's beta coefficient. It is

written as fa.

With this in hand, substituting into that In doing so, we are risky asset when full

we can make a final interpretation of Equation (7.23) by equation the regression coefficient we have just defined. defining the market's required rate of return on the yth general equilibrium conditions are satisfied:

We conclude from Equation (7.24) that the market's required rate of return on the 7th risky asset, for example the shares of common stock of theyth firm in the risky asset set, will be equal to the risk-free rate of interest plus a risk premium. That risk premium is equal to the asset's beta coefficient multiplied by the market risk premium, or the excess of the return on the market portfolio over the risk-free rate. If the first RF on the right-hand side of Equation (7.24) is transposed to the left-hand side, it follows that in market equilibrium the risk premium on the 7th asset will be equal to that asset's beta coefficient muliplied by the market risk premium. This result can be further interpreted with the aid of Figure 7.4. Consider the principal result of the theory described in Equation (7.24). If we envisage the market rate of return, E(RM), on the left-hand side of that equation, it follows that the "market beta" must be equal to 1. We depict this as point M in Figure 7.4. This figure shows the equilibrium relationship that

7 Asset markets and cost of money capital

147

exists between the beta coefficient and the expected rate of return on a security. Such a relationship, which can be referred to as the Security Market Line, must be linear, as will be clear from an inspection of Equation (7.24). We have, by definition of the underlying relations, two available points on this line: (i) the coordinates of M described by the expected return on the market portfolio and the value of (3 equal to 1 and (ii) the risk-free rate RF. We are therefore able to describe the locus. This Security Market Line implies that the beta coefficient now appears as a fundamental measure of the asset risk. This result may be further interpreted as follows. We have seen that by combining securities in a portfolio, and by taking advantage of the form of the covariance matrix that describes the risk relation between securities, it is possible to diversify in such a way and to such an extent as to reduce the portfolio risk to a minimum attainable level. But there is, of course, a limit to that attainable risk reduction. The proportion of a security's total risk that can be diversified away, or eliminated by diversification, is frequently referred to as unique risk or unsystematic or residual risk. This implies that there remains a systematic risk, also referred to as market risk, that cannot be diversified away. It is this remaining systematic risk, reflecting the manner in which a security's return fluctuates in relation to the market rate of return, that is captured and described by the security's beta coefficient. It follows that securities whose risk-return profile places them to the northeast of M on the Security Market Line in Figure 7.4 are more risky than the market portfolio, and those to the southwest of M are less risky. These relationships will be seen to be directly relevant to the specification by the asset market equilibrium theory of a firm's cost of capital. It follows from the preceding development also that the beta coefficient of a portfolio of securities will be a weighted average of the beta coefficients of the securities in the portfolio, where the weights are described by the proportions of the total portfolio value represented by each security. Nonstandard results of the equilibrium asset pricing theory It is beyond our present objectives to consider the large number of critical amendments that have been made to, and the advances that have been made beyond, the standard form of the capital asset pricing model that we have considered in the foregoing [see the extensive bibliographies in Elton and Gruber (1984)]. But a minimal comment will be made on two points. Consider again the basic result of the theory presented in j

j

-

R

F

]

(7.25)

This result, it will be recalled, was derived on the assumption that unlimited

148

II The neoclassical tradition

o(R)

Figure 7.5 borrowing and lending opportunities at a risk-free rate were available to investors. This is reflected in the first term on the right-hand side of Equation (7.25). If, however, no such borrowing and lending opportunities are available, the analysis can be cast in an alternative form that yields comparable results for equilibrium security prices and yields, and therefore for a firm's cost of capital (see Black, 1972; Elton and Gruber, 1984, Ch. 12). We restate the meaning of the Security Market Line in Figure 7.4 by observing that it passes through the two points of (i) the market portfolio point M and (ii) the rate-of-return ordinate corresponding to an asset or portfolio whose beta coefficient is zero. We can envisage the available rate of return on such a "zero-beta" portfolio and ask what determines its magnitude. Let us approach this question by considering Figure 7.5. The location of the market portfolio, or the optimum portfolio of risky securities, was observed to depend on the location of the risk-free rate at which borrowing and lending were assumed to take place. Let us then observe the actual market portfolio and note its risk-return profile and its location as described by point M in Figure 7.5. It is then possible to define, by making use of the linear locus tangential to M, the ordinate R'F. This can be interpreted as the risk-free rate such that if investors could borrow and lend at that rate, they would hold the market portfolio. Such a rate and such an opportunity as represented by R'F do not, of course, under the present assumptions, exist. But there do exist portfolios that provide a prospective rate of return equal to R'F. If RF were available, the expected return on the/th risky asset could, in accordance with

7 Asset markets and cost of money capital

149

our previous results, be defined as -R'F]

(7.26)

Consider then an asset such that its rate of return E(Rj) is equal to R'F. In that case, the value of the final term on the right-hand side of Equation (7.26) would have to be zero. This implies that the beta coefficient of any such asset or portfolio must be zero. We may therefore define RF as the equilibrium market rate of return on the zero-beta portfolios that could be chosen from the asset opportunity set. Such portfolios, lying within the opportunity set, are described by the ZC portion of the line RFZC in Figure 7.5. The equilibrium condition where borrowing and lending opportunities at a risk-free rate are not available can therefore be described by replacing the risk-free rate in Equation (7.25) by the rate available on a zero-beta portfolio, Rz. The equilibrium required rate of return on the 7th risky asset then becomes E(Rj)=Rz + pj[E(Rm) -Rz]

(7.27)

Given this result, the analysis and application to the firm's cost of capital can proceed as before. We shall return to the detailed specification of that cost of capital below. A final point of amendment to, or rather an extension of equilibrium asset theory beyond, the standard form of the capital asset pricing model can be noted without extended analysis (see Ross, 1976, 1977, 1978). This development, referred to as the Arbitrage Pricing Theory (APT), explains that in full generalized asset market equilibrium the expected return on a risky asset can be related to a number of fundamental "factors" as expressed in E(Rj) = a + bxh + b2l2 + b3l3 + -'+bnIn

(7.28)

where the //, / = 1, 2, . . . , n, refer to the factors or indexes on which the return on a risky security depends. The APT in fact bypasses the problem and theory of efficient portfolio construction we have examined in this chapter. It looks directly to the determinants of the asset's price and effective rate of return. The theory argues that if the existing risk premium on an asset is lower than is implied by the relationship envisaged in Equation (7.28), the asset can be considered to be overpriced. Investors would then sell the asset and engage in an "arbitrage" action by buying an alternative asset to replace it in their portfolios. Similarly, the asset would be purchased if its available risk premium were higher than the norm specified in Equation (7.28), and other assets would be sold in the rush to purchase it. This arbitrage activity in the risky asset market would, it is presumed, maintain market equilibrium and permit a direct approach to be made to estimating the asset's equilibrium rate of return. This again could then be incorporated into the analysis of the firm's cost of money capital.

150

II The neoclassical tradition

A utility-theoretic interpretation

The foregoing development has concentrated on the means and variances of the risky asset opportunities, and as such it exposes the essence of what we have referred to as the mean-variance theory of asset portfolio optimization. We can therefore exploit an alternative way of integrating this body of theory with the mean-variance utility functions we examined in the preceding chapter. The investor's mean-variance utility function may be written in the form U= U[E(RP), <J(RP)]. Its first-order optimization condition with respect to the yth asset in the investor's portfolio, where Xj now represents the proportion of his total wealth placed in the 7th risky asset, is

J£_ dE(Rp)

dE(Rp)

dU

dxj

da(Rp)

da(Rp)_ dXj

K

}

It follows that dE(Rp)_ dxj

_dUldor{Rp)

da(Rp)

dU/dE(Rp)

dxj

(7.30)

The first term on the right-hand side of Equation (7.30), the negative of the ratio of the partial derivatives of the utility function, is recognizable as the investor's marginal rate of substitution between risk and return. This must be equal in equilibrium to the market price of risk or the slope of the Capital Market Line. Using previous notation, we may make use of the market price of risk at the point of utility optimization, and the condition described in Equation (7.30) may be written as dE(Rp)_E(RM)-RF dXj

dcr(Rp)

(J{RM)

dXj

The investor's overall wealth allocation problem is now solved by placing a proportion of his wealth equal to ^Xj in a risky asset portfolio and the balance, 1 — 2*,-, in a risk-free asset yielding a return as before of RF. The investor's expected return and risk exposure then become E(RP) = ^XJE(RJ)

+ (1 - ^Xj)RF

(7.32)

and

o^) = [22wvj' / 2

(7-33)

Differentiation of Equations (7.32) and (7.33) with respect to Xj provides dE(Rp) _ = E(Rj)-RF

(7.34)

7 Asset markets and cost of money capital

151

and d(T(Rp)

^Xtdij

(7.35)

a(RM)

We recall, in the final term of Equation (7.35), that the portfolio on which interest can be focused when equilibrium conditions are satisfied is once again the market portfolio. Equations (7.34) and (7.35) may then be substituted into the utility optimization condition in Equation (7.31) to provide (7.36) In the final term of this equation, the risk of the jth asset in the portfolio is recognizable from the result established in Equation (7.8). From Equation (7.36), using Equation (7.9), E(Rj)-RF=[E(RM)-RF]

Cov

<%*">

a

{K)

(7.37)

This states again that when general equilibrium conditions are satisfied, the market's required risk premium on theyth risky asset will be a linear function of the risk premium on the market portfolio. The final term on the right-hand side of Equation (7.37) again incorporates they'th asset's beta coefficient. Wealth allocation separation theorem We recall that from the point of view of the individual investor the financial asset market is assumed to satisfy the requirements of a perfectly competitive market. We assume that conditions in the market are such that the optimum portfolio of risky assets has been determined at point M in Figure 7.2, reproduced as point M in Figure 7.6. It then follows that the investor's decision as to the division of his wealth endowment between this risky portfolio and the risk-free asset will depend on the form of his utility function, in particular on the degree of risk aversion it exhibits. If, for example, the form of the indifference curves is such that the relevant point of tangency is established at M in Figure 7.6, the individual would place his entire wealth in the risky portfolio. If, on the other hand, the utility function exhibited a higher degree of risk aversion and established tangency at point 5, the investor would divide his wealth between the risky portfolio and the risk-free asset. The rate of return on his total wealth position in that case, and the overall risk to which he would be exposed, would be described by the intercepts on the axes (not shown in

152

II The neoclassical tradition E(RP)

o(Rp)

Figure 7.6 the figure) adjacent to S. If the investor, instead of investing in the risk-free asset and thereby lending at the given interest rate RF, were to borrow at that rate and invest the proceeds in the risky portfolio, he might optimize at point T on the RFMZ opportunity line. This could occur if his utility function exhibited a somewhat lower degree of risk aversion. The investor would then be increasing his expected return above that obtainable on the market portfolio M, but he would be increasing the degree of risk to which he was exposed. His expected return and risk are indicated by the intercepts on the coordinate axes adjacent to point T. This, therefore, establishes the wealth allocation separation theorem. The investor's final decision regarding the distribution of his wealth endowment is separable from the decision as to the structure of the optimum portfolio of risky assets. The result is dependent on the assumption that all investors can borrow or lend any desired amount at an exogenously given risk-free rate of interest in perfect money capital markets. But a number of difficulties arise. The assumption of perfection in the asset market may not provide a reasonable basis from which to begin an analysis of the trading that takes place in it. There may be quasi-monopsonistic and monopolistic phenomena on the respective sides of the market, and neither transactions nor the gathering of information may be costless. Furthermore, the risk-free rate is likely to be unstable. To the extent, also, that trading takes place in the market at nonequilibrium prices, the windfall gains and losses that occur can be expected to have endowment-redistribution and expectations-inducing effects. Finally,

7 Asset markets and cost of money capital

153

the supply of financial assets, for example the debt and equity capital securities of firms, is not likely to be stable, as firms change, grow, merge, expand, decline, and die. But the final results of the orthodox theory, at least in the form in which we have exhibited it, are such that when equilibrium conditions are satisfied, three situations obtain. First, and assuming we can rule out a corner solution in which the investor's entire wealth would be placed in the risk-free asset, all investors will hold the same portfolio of risky assets. Some of every asset will be held by every investor. Moreover, there would be no reason, in the idealized world of the general equilibrium asset theory, for an individual to retain sole proprietorship of a firm. He could reach a higher utility level by exchanging part, perhaps most, of his ownership for other components of the market portfolio and realizing the benefits of asset diversification. These results, of course, are demonstrably false as a description of the world. The whole advantage of the theory, therefore, as claimed by adherents to the equilibrium-theoretic tradition, rests in the extent to which the results generated by the abstractions contained within it provide reasonable and usable approximations to the rates of return actually observed on risky securities. Second, the proportionate distribution of investor's wealth will depend on the degree of risk aversion in their assumed mean-variance utility functions. Third, all investors will exhibit the same subjective marginal rate of substitution between risk and return, equal to the slope of the objectively determined linearized market opportunity locus, or the objectively attainable marginal rate of transformation between risk and return. This may be further conceptualized as the market price of risk. The cost of money capital The foregoing analysis implies that the measure for which we have been searching, the specification of the firm's cost of equity capital or of p in the economic valuation equation, is now defined by the asset market's equilibrium required rate of return on the firm's equity capital security. It is defined precisely in Equation (7.24). We should note carefully, however, that this result and this specification follow only so long as we are committed to the presumed validity and relevance of the asset market equilibrium theory from which it was derived. The principal result of the theory, and therefore the principal implication for the firm's cost of money capital, has to do with the nature of the covariation between the rate of return on the risky asset, in this case the common stock of the 7th firm, and the rate of return being earned on the market portfolio. The analysis is concerned with what we designated at an earlier stage as the "variability risk" in the income stream generated by the risky asset.

154

II The neoclassical tradition

The total variability of an asset's rate of return can be separated into what we have called "systematic risk," or "market risk," on the one hand and "unsystematic," or "unique," risk on the other. The latter can be diversified away by an investor who combines the asset with other assets in a portfolio. But the market or systematic risk cannot be diversified away, for it remains by virtue of the fact that asset returns move together with the asset market in general, being linked to the market fluctuations by the asset's beta coefficient. We need to consider more precisely, however, what it is that contributes to or determines the overall variability of equity capital rates of return. We recall the argument in Chapter 2 regarding the leverage structures of the firm and the manner in which the operating and financing structures contribute to the instability of the firm's residual income stream. That overall instability, we saw, was due to both external forces and economic developments bearing on the firm and to internal structural forces that determined the manner in which the firm's income stream behaved in reaction to external events. The integrated view of the firm's production, real capital investment, and financing structures that we examined in Chapters 2 and 4 enables us to isolate for analytical purposes what we might call a first-stage or operational leverage effect. Given that variations occur in the general level of economic activity, this will cause fluctuations in a firm's sales revenue. But then, in reaction to those developments, magnified fluctuations will occur in the firm's net operating income stream. The extent to which this will happen will depend on the incidence of fixed operating costs that have to be deducted from gross revenue before the net operating income is determined. Such fixed operating costs stem, we have seen, from the decision to employ durable real capital assets in the firm. The resulting instability at the operating income level has been referred to as a reflection of operational leverage. This instability or risk may be referred to as "business risk" or "operating risk.'' In speaking of business risk, we are envisaging the character or quality of the operating income stream that is generated by the total asset investment at work in the firm. We saw, in the examples we gave in Chapter 2, that two firms of the same size, in the same industry, with the same total asset investment, and subject to the same variations of economic fortunes may have adopted different financing methods. They may have different debt-to-equity ratios in their financing mix or in the structure of the liabilities sides of their balance sheets. The possibility of different financing methods gave rise to different possible degrees of financial leverage and led, as a result, to different potential degrees of instability in the firm's residual income stream. The instability in a firm's residual income stream, or in the rate of return earned on its residual equity, is thus due to (i) general external economic fluctuations, (ii) the firm's operating structure, and (iii) the firm's financing structure. We can expect that the overall risk in the firm's rate of return on

7 Asset markets and cost of money capital 155 equity will depend on the force of, and the interaction between, these several sources of instability. We can expect the residual earnings of firms with different operating and financing structures to exhibit different degrees of relationship with the rate of return earned on the asset market as a whole. In other words, firms with different operating and financing structures can be expected to exhibit different beta coefficients. This points to a fuller specification of the functions determining the required rates of return on a firm's debt and equity securities, or its costs of debt and equity capital. Let us designate the business or operating risk in the firm by the symbol 6 and the further degree of risk induced by financial leverage as 8. We shall employ the symbol TMC to refer to the total money capital employed by the firm. We can then write the cost of debt and equity capital, respectively, as r = r(0, 6, TMC)

(7.38)

p = fK0, 5, TMC)

(7.39)

The total money capital variable is included here as a size variable to capture what might be economies of scale in the use of the financial markets, even though in the analysis in this chapter, under the assumption of perfect money capital markets, no such economies of scale apply. In our discussion of the firm's constrained optimization in Chapter 4, we referred to these debt and equity costs as r = r(K, D) and p = p(K} D), concentrating on the firm's money capital decision variables. But that constrained optimization analysis did take extensive account of the determinants of the firm's operating structure, and it is necessary to bear in mind also, therefore, the impact of such structural decisions on the money capital financing costs we now envisage. We shall return to the question of the cost of money capital in the next chapter.

CHAPTER 8

The cost of money capital: further analysis and controversy One of the principal conclusions we reached in Chapter 4 was related to the description of the firm's cost of money capital. We specified there the full marginal costs of debt and equity capital, and these were related to the full marginal cost of relaxing the money capital availability constraint. Equation (4.47) in Chapter 4 instanced an equilibrium or an optimization relation between these full marginal costs. We begin our discussion of the cost of capital in this chapter by considering those concepts further. In doing so, we adopt a quite different analytical stance from that of the preceding equilibrium asset market theory. There we examined the implications of general equilibrium in perfect financial markets. In what follows, we shall be initially concerned with questions of partial equilibrium in conceivably imperfect markets. We shall return to some general equilibrium considerations in later sections of this chapter. Illustration of the full marginal cost of relaxing the money capital availability constraint The following example illustrates the empirical meaning of the full marginal cost of increasing the amount of money capital available to the firm by raising an increment of debt capital. It provides an instance of the concept first introduced in Equation (4.31). It will be useful to keep in mind throughout the discussion the fundamental definition of the firm's cost of money capital. Definition The cost of money capital is the minimum required rate of return that must be earned on investing marginal money capital in the firm in order not to cause a dilution of the owners' economic wealth position. Let us suppose that a firm, with a given total asset investment and a given debt-to-equity financing mix, is generating a residual income stream that provides the common stockholders with earnings of $3 per share. If the owners' required rate of return is 10 percent, the market value of the shares of common stock will be equal to $3 capitalized at 10 percent, or $30. Suppose now that 156

8 Cost of money capital: further analysis

157

the firm decides to undertake an investment project that is financed by a new issue of debt capital. This issue of debt will increase the degree of leverage in the firm's financing structure, and as a result of their increased risk exposure the owners' required rate of return will be assumed to rise from 10 to 12.5 percent. If there were no change in the residual earnings per share, the market value of the shares would fall from $30 to the equivalent of $3 capitalized at 12.5 percent, or $24. This reduction in market value, or in economic value, is referred to as a dilution of the owners' economic position. This could be expected to happen if the project in which the new debt capital is invested generated only sufficient earnings to pay the interest on the debt. In order to prevent this dilution, the new debt capital, when it is put to work in the firm, must generate a level of earnings greater than the cost of the interest on the debt. After the new investment and financing has occurred, the residual earnings must increase by such an amount that when the new earnings per share are capitalized at the new higher capitalization rate, the value of the common stock will be at least as high as it was before. If, as in the example, the capitalization rate rose from 10 to 12.5 percent, the earnings per share would need to rise from $3 to $3.75 in order to prevent dilution. The following example will explore this set of relationships more fully. The firm's constrained objective function, analogous to Equation (4.28), may be stated as

xh K, D

PK

'

A(Q)f(X)afXr(KD)D](lt) +\[K + D-b'X]

}

(8.1)

where/(Z) describes the production function of the firm in vector notation, X describing an n-element factor input vector. Similarly, a' refers to a factor cost vector (meaning by this the operating flow cost as previously examined), and b' describes a vector of money capital requirement coefficients. In Equation (8.1), the firm's net income, contained in the first square brackets on the right-hand side, is multiplied by the complement of the tax rate, r, to define the residual net after-tax income actually available to the owners. The remaining variables are defined as previously. To trace out the effects of marginal debt financing, we examine the partial derivative of Equation (8.1) with respect to the debt variable D: d _ 7r(l-t)dp 2 6>D~ p dD It follows from Equation (8.2) that

158

II The neoclassical tradition

Dividing throughout by 1 — t yields

where X' equals the marginal productivity measure multiplied by a tax stepup factor to express it on a before-tax basis. The tax step-up factor, which also appears on the left-hand side of the equation, is the reciprocal of 1 minus the firm's tax rate. Both sides of the optimization condition in Equation (8.4) are expressing the firm's marginal cost of debt capital, and the left-hand side exhibits in more detail the "full marginal cost of relaxing the money capital availability constraint." The first two terms taken together describe the derivative of the firm's interest burden with respect to debt, d[r(K, D)]/dD, or the increase in the interest burden that results from the increased use of debt. We refer to this magnitude (r + D dr/dD) as the firm's marginal direct cost of debt. Recognizing the first term, r, as the interest rate that has to be paid on the marginal debt itself, the marginal direct cost of debt will be greater than the interest cost of the marginal debt. This is because the higher interest rate on debt capital that results from the increase in the debt-to-equity financing ratio will have to be paid, not only on the new debt now being issued, but also on the previously existing debt when that matures and has to be refinanced. Recalling the foregoing discussion of the equity owners' position, the minimum rate of return that must be earned on the marginal debt capital must exceed this marginal direct cost of debt by an amount sufficient to offset any tendency to a dilution of the market value of the equity. Consider, therefore, the following example. The magnitudes are hypothetical and no doubt exaggerated, but they exhibit the nature of the causation involved. Employing a time subscript of zero to refer to the firm's situation before the new financing and investment and subscript 1 to refer to the situation after the new financing, let us assume the following situation: D o = $3 million (the market value of debt capital) Vo — $6 million (the market value of equity) WO = DQ + VO = $9 million (the total market value of the firm's capital securities) ro-0.05 DI0 = $150,000 (debt interest) p 0 = 0.10 Number of equity shares outstanding = 100,000 Earnings per share (EPSo) = $6 Market value per share (So) = $6/0.10 = $60

8 Cost of money capital: further analysis

159

Table 8.1 Income statement data ($000)

Operating income Less debt interest Earnings before tax Less tax

Before new financing

Required position after new financing

Incremental data

$1,350 150

$1,800 300

$450 150

1,200 600

1,500 750

300 150

Net income (Residual equity earnings)

600

750

Earnings per share

$6

$7.5

Market value per share

$6/0.10 = $60

$7.5/0.125 = $60

150 $1.5

Marginal debt financing (AD) — $2 million ri = 0.06 DIi = $300,000 (after refinancing all maturing debt) P!= 0.125 ^ = ^=0.50 In this situation, the full marginal cost of debt as defined on the left-hand side of Equation (8.4) quantifies as ^

^

J

3

^

r

0.06 + 0.015 + 0.15 = 0.225

(8.5)

The full marginal cost of debt, or the required rate of return on a before-tax basis necessary to establish economic viability, is 22.5 percent. To reinforce the understanding of these relationships, consider the pro forma income statement of the firm in Table 8.1. The data in Table 8.1 imply that the residual earnings per share of equity after tax would increase by $1.50, sufficient to increase earnings per share to $7.50. When this amount is capitalized at the new higher capitalization rate of 12.5 percent, the market value per share of $60 can be maintained, and no equity dilution will have occurred. But this will occur only if the operating income of the firm increased by $450,000, or by 22.5 percent of the new $2

160 II The neoclassical tradition million investment, precisely the gross before-tax required rate of return or the full marginal cost of the debt as specified in Equation (8.5). Weighted average cost of capital Leaving aside now the question of computation, and ignoring the technical adjustment for the firm's tax rate, we recall the firm's full marginal cost of equity that was brought into relation with the full marginal cost of debt in Equation (4.47). At the firm's optimum structural planning point, both these full marginal costs were equal to the magnitude p \ , which therefore emerged as the firm's optimum planning discount factor or valuation rate. We consider now whether any traceable relation exists between this planning discount factor and the more familiar concept of the firm's weighted average cost of capital. It will be shown that the two are in fact equal to one another when the firm has been brought to its optimum size and structure. At that point, and when all parts of the interdependent planning nexus have been brought to their optimum solution values, the weighted average cost of capital will be equal to both the full marginal cost of debt and the full marginal cost of equity. It can therefore be employed for valuation purposes at the margin. This will, in fact, provide a potential linkage with the results of the general equilibrium-theoretic analysis. But it follows from the full set of optimum conditions that this is admissible only when the firm has chosen its mutually determined optimum levels of output, factor usages, asset investments, and financing sources. The reasons for this will be clarified in the following argument. Consider the following notation: V, IT, and p refer, as before, to the market value, profits, and capitalization rate applicable to the owners' equity, and D, F, and r refer to the market value, interest earnings, and market yield (average rate of interest) on the debt capital of the firm. In the planning model of the firm with which we are working, the market value and the book value of the debt will be equal. This is because we are supposing that following any change in the amount of debt, or in the average rate of interest on the debt, all of the firm's previously existing debt, as in the foregoing numerical illustration, is refinanced at the new rate of interest. Finally, W and O refer to the total capitalized value, W, of the net operating income of the firm, O, and the relationship O/W will provide a measure of /, the effective rate of capitalization at which the total earnings of the firm are being valued in W. The total value of W, moreover, will be equal to the sum of D and V. These definitions imply the following relations. V =7T/p

7T=pV

(8.6)

D=F/r

F=rD

(8.7)

8 Cost of money capital: further analysis

W = O/i

O=iW

161

(8.8)

If, at any planning point, a set of security values and income statement items are observed and described as in the foregoing list, the following definitional statements can be made. l

(89)

-w~~ttT

whence

From Equation (8.10), the overall capitalization rate, /, which may now be referred to as the average cost of capital, is seen to be a weighted average of the costs of debt and equity capital, r and p, respectively. Each such capital cost is weighted by the relative importance of the corresponding capital in the total market value of the firm's securities. In the optimization model, the debt and equity capital costs were assumed to be functionally related to the amounts of debt and equity capital measured at book values (as book values represent the decision variables confronting the firm1). Equation (8.10) may therefore be written as i = r(K,D)^

+ p(K,D)^

(8.11)

We are interested now in the combination of debt and equity capital at work in the firm, D and K, that will afford the minimum attainable value of / when the firm is at its optimum capital usage point. We accordingly differentiate Equation (8.11) partially with respect to D and K, setting the partial derivatives equal to zero, and noting that W-D + V. We note also that because we are investigating these relationships at the optimum planning point, we are at the situation at which, as seen in Equation (4.51), dVldK is equal to unity. Employing this fact, dW/dK may be replaced in what follows by BW/dV. Differentiation of Equation (8.11) yields

1

An additional reason for using book values rather than market values as functional arguments is that the use of market values introduces a bias due to the fact that they reflect business risk (on which we shall comment further later) as well as financial risk. This implies that market values may vary, quite independently of financial risks, in reaction to changes in business risk, as the latter induces changes in the debt and equity required rates of return or capitalization rates. See Barges (1963), Turnovsky (1970, p. 1065), and Herendeen (1975, p. 125).

162

II The neoclassical tradition

Ddr

di

D , Vdp

Multiplying Equation (8.12) by W and rearranging yields

r+D

w+v^=4+pw

(814)

Similarly, Equation (8.13) yields

Equations (8.14) and (8.15) together imply that when full optimization conditions are satisfied, the full marginal cost of borrowing and the full marginal cost of equity are both equal to the weighted average cost of capital. It is instructive to compare the results in Equations (8.14) and (8.15) with Equations (4.47) and (4.48). Combining the equations, it follows that r
(8.16)

The relationships we previously hypothesized as relevant to the optimum structure of the firm continue to hold. The average rate of interest on the debt is less than the weighted average cost of capital (which, at the structural optimum, is equal to the full marginal cost of relaxing the money capital availability constraint), and this is in turn less than the cost of equity capital. Moreover, it follows from Equation (8.16) that \ =-
(8.17)

a result corresponding to that reached in Equation (4.48).

The question of business risk We have defined business risk in terms of the degree of instability in the firm's net operating income. It might be measured by the standard deviation of the probability distribution of that operating income, or, to establish a "risk class" of firms of different size, by the coefficient of variation of that distribution (see Modigliani and Miller, 1958). If the firm did not employ any debt capital, all of the net operating income would become the property of the equity holders, and the measure of business risk would in that case be synonymous with the comparable dispersion parameter of the distribution of equity earnings, or, in some formulations, earnings per share of common stock. But in the presence of debt financing the concept of business risk must be referred consistently to the operating income. That, as we have noted, provides an indi-

8 Cost of money capital: further analysis

163

cation of the income-generating ability of the total asset investment at work in the firm, irrespective of how that investment has been financed.2 We may assume that both the suppliers of debt capital and the residual equity holders are risk averse and that the respective cost of money capital functions contain business and financial risk arguments in the following general form3: r = r(0,0L)

rl>0,r2>0

(8.18)

p = p(0,dL)

Pl>0,p2>0

(8.19)

Equation (8.18) states that the rate of interest on debt capital depends directly on the business risk, here referred to as 0, and that it is directly related also to the debt-to-equity, or financial leverage, ratio, here described as L = D/K. Not only does the interest cost increase with financial leverage, however, but also its positive response to that variable increases with the degree of business risk. Notationally, r\ on the right-hand side of Equation (8.18) refers to the partial derivative of the interest cost function with respect to the first variable in the function. Similarly, r2 refers to the partial derivative with respect to the second variable. Comparable notation is employed in conjunction with the equity cost function in Equation (8.19). The equity cost function may be interpreted in terms analogous to those implicit in the specification of the cost of debt capital. If money capital is invested in the firm in such a way as to increase the degree of business risk, both the interest rate on debt capital and the equity owners' required rate of return can be expected to increase. The investment, therefore, must generate an income stream that is somewhat larger than that necessary to pay simply the interest cost of the marginal debt capital that might have been used in the financing. We saw earlier in this chapter that in order to specify the full marginal cost of debt, account needed to be taken of (i) the additional earnings necessary to provide for the incremental interest costs when previously existing debt has to be refinanced at a new higher interest rate (this was specified as D dr/dD) and (ii) the further additional earnings necessary to increase the residual equity owners' income sufficiently to offset any tendency for their economic position to be diluted; this was specified as V dpi3D. Similarly, we now have to bear in mind the impact on 2

3

See Turnovsky (1970), where business risk is defined in terms of the variability of the net equity earnings even when those earnings reflect a deduction for interest payments on debt capital and even though, as a result, the variability of the residual equity earnings will in that case reflect instabilities due to financial risk as well as business risk. Turnovsky's paper is an important discussion of the issues raised by the original Vickers model and it throws significant light on some of the comparative static properties of the argument. This form is suggested by Turnovsky (1970), though, as indicated, tne business risk argument here refers to the dispersion of the net operating income of the firm, not the residual equity income as Turnovsky envisages.

164

II The neoclassical tradition

the firm's internal required rate of return on investment in income-generating assets due to the influence of dr/dO and dp/dd. Moreover, not only will the incidence of business risk change the money capital costs in the manner we have just observed but also it opens up a possible relation between the degree of business risk and the degree of financial leverage, or the optimum financing mix, in the firm. Firms that confront a high degree of business risk and employ a high degree of operational leverage may be reluctant to increase the residual owners' risk further by adding to that operational leverage a significant degree of financial leverage. Modigliani-Miller hypotheses: weighted average cost of capital revisited We have observed that the firm's weighted average cost of capital is usable as an investment decision criterion only when the firm has been brought to an optimum structural position. It is not valid or reliable at infra-optimum planning stages. But the weighted average concept has gained wide currency in the finance literature and in business practice. This has been due in large part to the seminal arguments of Modigliani and Miller, who have demonstrated that under well-specified assumptions and for a specified risk class of firms the weighted average cost of capital is invariant with respect to the firm's financing mix. If, of course, such an average cost of money capital is invariant, its marginal value is equal to the average value, and it is thereby established as a valid decision criterion. The traditional theory of finance, on the other hand, argues that the assumptions necessary to establish the Modigliani-Miller invariance cannot be expected to hold in fact, and an alternative construction of the money capital cost theory is presented. For purposes of the following analysis, recall the definitions contained in Equations (8.6)-(8.8). From these definitions, the weighted average cost of capital was derived in Equation (8.10). The notional capitalization rate defined there as /, the effective rate at which the total income stream of the firm is capitalized in the total market value of the firm's capital securities, is not an observable variable in the financial asset or money capital market. The data that are observable are the r and the p capitalization rates, depending on market asset values and implicit required rates of return in the debt and equity sectors of the money capital market. Consider the equity capitalization rate, p:

(820) From this it follows that p = i + (i-r)D/V

(8.21)

8 Cost of money capital: further analysis 165 This appears to present us with the proposition that the equity capitalization rate is a linear function of the debt-to-equity financing ratio, DIV, in the firm. In fact, Equations (8.10) and (8.21) contain the two principal propositions in capital financing theory that were presented in the celebrated Modigliani and Miller paper and have become the source of a very extensive debate. 4 But care must be taken in the interpretation of the average capital cost and the equity capitalization rate formulas derived in these equations. These expressions are merely definitional statements implicit in the structure of things observable at a given moment in time. They do not, as they stand, provide a theory of the cost of capital or of the owners' required rate of return. To move from the definitional propositions to a theory of money capital costs, it is necessary to introduce a set of behavior postulates to describe the ways in which participants in the financial markets actually behave in conditions of change. Or we may put this important point in other terms. So far as the Modigliani-Miller theorems specify conclusions similar to those in our definitional Equations (8.10) and (8.21), they are, in their framework of analysis, deduced as equilibrium conditions; and when equilibrium conditions are satisfied, no further change or movement with the objective of improving one's economic position is possible. We must therefore avoid any statement about the manner in which a firm may be able to move along the apparent functional relation described, for example, in Equation (8.21). The equation serves only to identify the relation that would exist if conditions of equilibrium in the financial asset market were also assumed to exist. If we can assume that perfect markets exist, that no transactions costs or taxes are incurred, and that both firms and individuals can borrow at an externally given market rate of interest, the following argument regarding a possible market equilibrating activity can be posited.5 4

5

Modigliani and Miller (1958). See also, for an early but highly valuable summary of the debate, Robichek and Myers (1965) and, for an up-to-date view, Brealey and Myers (1984, p. 357f., and references cited there, especially Durand, 1952; Hamada, 1969; and Fama, 1978). The paper by Hamada, which we shall refer to again below, demonstrates the consistency between the Modigliani-Miller propositions and the results of the financial asset market theory that we examined in Chapter 7. The assumptions invoked in this sentence should be carefully noted, as the violation of any one or more of them will be sufficient to invalidate the Modigliani-Miller equilibrium conclusions that will be deduced in what follows in the text. For a discussion of transactions costs and market imperfections, see Baumol and Malkiel (1967). See also the highly insightful discussion of the Modigliani-Miller theorem in Nickell (1978, Ch. 8), where a U-shaped average cost of capital curve is derived similar to that described in Figure 8.2 below (Nickell, 1978, p. 177). Moreover, Nickell demonstrates the contribution to the shape of this cost of capital curve made by the risk of bankruptcy, an issue we have not examined. As to the general perspective we have adopted on the money capital markets, Nickell observes that "the assumption of 'perfection' for a capital market in a world of uncertainty is clearly liable to lead to absurdity. . . . Capital markets are inherently imperfect and . . . these imperfections are nontrivial in their effects" (p. 167). The upshot of Nickell's analysis is to confiim the fundamental proposition we examined in a preliminary fashion in Chapter 4 and to which we shall return in Part III,

166

II The neoclassical tradition

Table 8.2 Firm A

Firmfl

Total assets Internal rate of return Net operating income

$10,000 10% 1,000

$10,000 10% 1,000

Market value of equity Market value of debt

10,000

6,000 5,000

10,000

11,000

Total market value of firm's securities

Let it be supposed that two firms, A and B, of the same size and operating in the same industry in the same economic environment both earn the same rate of return on the total capital invested and that they both exhibit the same degree of risk in their operating income streams. They differ only in the financing mix they have adopted. We shall suppose that firm A has financed its asset investments completely from equity capital sources and that firm B has financed 50 percent of its requirements by means of debt capital. We shall assume that for both firms the internal rate of return on capital employed, or the ratio of the net operating income to the book value of the assets, is 10 percent. Finally, both the firm and its security holders can borrow at 4 percent, the total value of the assets is $10,000, and no account is taken for the present of either corporate or personal taxes. Our objective is to demonstrate that on the assumption of a smoothly functioning investor arbitrage mechanism in an assumedly perfect capital asset market, the total market value of the securities of the levered firm B cannot diverge permanently from the total market value of the unlevered firm A. If it is possible to substantiate the claim that the total market values of the respective firms' securities, W, must be equal, then for a given total income stream, O, the ratio of O/W must be the same for both firms. The weighted average cost of capital for both firms, or i = O/W, will then also be equal. It would be able to be said, then, that the weighted average cost of capital to the firm is invariant with respect to its financing mix. Suppose that a ''temporarily anomalous" situation exists in the financial asset market and that the securities of the levered firm B are overvalued relative to those of firm A. In that event a supposed arbitrage mechanism would correct the anomaly and restore the equality of market values. Table 8.2 indicates the sense in which firm B is assumed to be overvalued. namely, "the (firm's) financing decision and the investment decision are interdependent" (p. 178).

8 Cost of money capital: further analysis

167

Table 8.3 Investor alternative income positions Income from 10 percent ownership

Firm net operating income Less debt interest at 4% Residual equity income of which 10% equity income — Less interest on personal debt Net portfolio income

Firm A

Firm B

$1,000

$1,000 200

1,000 100 16

800 80

84

80

Because of the anomalously high market value of its equity, the total market value of the firm, $11,000, is greater than the comparable unlevered firm's value of $10,000. Let us imagine that an owner of 10 percent of the common stock of the relatively overvalued firm B can sell his shares and purchase a 10 percent ownership of the relatively undervalued unlevered firm A, or, that is, that he can "arbitrage" in the firms' common stock securities. Given that such an investor will obtain $600 from the sale of his 10 percent ownership in firm B and that he requires $1,000 to acquire the contemplated 10 percent ownership of firm A, he will need to borrow the difference of $400 on his personal account, conceivably against the collateral security of the equity stock of firm A that he is about to acquire. Such borrowing, it has already been assumed, can be made at the same rate of interest as that at which debt capital is available to the firms. Consider, then, the income accruing to the investor from his alternative positions (Table 8.3). Table 8.3 indicates that by selling his 10 percent ownership of the levered firm B and purchasing a corresponding proportionate ownership of the unlevered firm A, the investor can increase his income prospects from $80 to $84. But as the market arbitrage proceeds, the selling pressure can be expected to reduce the market value of the firm B security, and the corresponding buying pressure will increase the value of firm A. The motivation for arbitrage will disappear when the divergence of market values between firm A and firm B has been eliminated, and a common market value will then have established an equality of average capital costs. The operating income stream, O f being given for the two firms, and the F s having been brought into equality, the / = O/W must be the same for each of the firms. A good many reasons can be adduced why things may not work out as smoothly as we have just supposed. Doubts may be expressed as to the ability of individuals and firms to borrow on comparable terms, or as to whether the

168

II The neoclassical tradition

investor, by thus substituting ' 'home-made leverage" in his personal portfolio for what was previously a levered position by reason of the financial leverage in firm B's capital structure, has in fact maintained a strictly comparable risk exposure. The assumption of zero transactions costs and the assumed absence of corporate and capital gains taxes could also be questioned. But leaving these details aside, we can note two aspects of the argument that give it a kinship with the static equilibrium asset pricing theory presented in Chapter 7. These have to do with the assumption of an exogenously given and fixed market rate of interest and that of perfect financial market conditions. Additionally, if all firms in a given "risk class" exhibited the same average cost of capital, /, this would provide an empirical vindication of the analytical proposition that the financing cost is invariant with respect to the financing mix. A basis would thereby be provided for a reliance on the posited linear form of the equity cost function described in Equation (8.21). We take note at this point of the methodological difference between the foregoing and the so-called traditional way of looking at things. The latter, as we observed at the beginning of this chapter, takes a partial equilibrium approach under assumptions of conceivably imperfect markets rather than a general equilibrium approach under assumptions of perfect capital markets. We recall the two definitional equations in terms of which we are conducting the argument:

i=r +p

w w

(8 22)

-

and p= i+ (i-r)^

(8.23)

It may be claimed that the cost of equity capital described in Equation (8.23) is what it is because the weighted average cost of capital, /, as defined in Equation (8.22), is constant. But it might be said, on the contrary, that the empirically observable equity and debt costs provide the basis from which deduction about capital costs can proceed and that the average cost function is the conclusion to which we argue. Then the cost of a firm's debt capital will be whatever it is observed to be by virtue of the supply and demand conditions in the debt capital sector of the money capital market, taking account of the degree of risk aversion in the attitudes of the suppliers of debt capital funds. Similarly, the cost of equity capital will reflect conditions in the equity capital sector of the money capital market. The differences of viewpoint can be illustrated by the functional relations depicted in Figures 8.1 and 8.2. Figure 8.1 depicts the essence of what has become known as the "net

8 Cost of money capital: further analysis

169

= i + (i-r)D/V

D/V

Figure 8.1

P=pipiV)

D/V

Figure 8.2

operating income theory" of the firm's capital structure and money capital costs. It is also referred to as the "economic entity theory," and it reflects the Modigliani-Miller analysis. The theory states that the value of the firm, meaning by that the total market value of the firm's capital securities, is equal to the total net operating income stream capitalized at a capitalization rate equal to the weighted average cost of capital. It is defined by W in the preceding

170

II The neoclassical tradition

derivation, where W=O/i. As we have seen, for a given income stream, Oy and a constant capitalization rate or weighted average cost of capital, /, the economic value of the firm is uniquely determined as W. The capitalization rate, /, is constant for a given firm, or risk class of firms, independent of the degree of financial leverage in the capital structure. In Figure 8.1, the financial leverage ratio, defined as the ratio of debt to equity capital measured at market values, is shown on the horizontal axis. The cost of equity capital, derived in Equation (8.21) and reproduced in Equation (8.23), is a linear function of the debt-to-equity ratio, provided, as is assumed, that the rate of interest on the debt is also given and fixed. This way of looking at the money capital cost problem has been called the entity theory because it argues that the total economic value of the firm as an entity depends, in the manner we have seen, on the total income stream generated by the firm as a whole. It follows that if the rate of interest on the debt is given at r, and the market value of the debt therefore equals the debt interest payments capitalized at the rate of r percent, the value of the equity is simply the difference between the total entity value, W, and the value of the debt, D. On one view, the value of the equity is a residual derived from the relation V= W-D. On another view, the cost of equity capital is a derived residual. For given the value of /, the weighted average cost, and r, the cost of debt, the value of p is deduced in the manner of the preceding analysis. This direction of relationship or causation has been indicated in Figure 8.1 by the arrows, which are intended to depict the initial statements or suppositions from which the causation proceeds and the conclusion to which it leads. It should be borne in mind in evaluating this line of approach that the analytical postulate of the constancy of the overall capitalization rate, /, has been vindicated, assumedly, by the empirical demonstration of the market arbitrage mechanism that we discussed earlier in this chapter. Figure 8.2 presents the alternative approach to the money capital cost and capital structure problem. It is frequently referred to as the "traditional theory." The theory assumes, as shown in the debt cost function in the figure, that the rate of interest on debt will be determined by the market conditions in the debt capital sector of the money capital market. It is shown in the figure as an increasing function of the financial leverage ratio, and unlike the entity theory, it sees no reason to assume that the interest rate on the debt will be constant. Similarly, the cost of equity capital is depicted in Figure 8.2 as depending on conditions in the equity capital sector of the money capital market and is shown as an increasing function of the financial leverage ratio. No necessary, or a priori, form can be prescribed for this function, which, like the debt cost function, is here an approximation to the fuller specification that takes account of the business as well as the financial risks in the firm. Moreover, the form

8 Cost of money capital: further analysis

171

of the equity cost function will depend on the degree of risk aversion inherent in the attitudes of the suppliers of equity capital funds. If it is argued that both the debt cost and the equity cost are determined by money capital market conditions, it follows that the weighted average cost of capital will be, in fact, the weighted average of these separately determined debt and equity costs. Again the arrows in Figure 8.2 are designed to indicate the initial statements or conditions from which the relationship proceeds and the conclusion to which it leads. While, on one view, then, this approach argues from the debt and equity capital costs to the weighted average cost of capital, on another view it argues from the observation of the market value of the debt, D, and the market value of the equity, V, to the total market value of the firm's securities, W. Against the entity theory's V=W — D, we can therefore set the traditional theory's

Adjustment for the firm's tax liability In an earlier section of this chapter an adjustment was made to the firm's constrained objective function to acknowledge the effect of the liability to taxation on residual net income. The entity theory accomplishes an adjustment for taxes in the following manner. Using established notation, the tax deductibility of interest on debt capital implies that 7r=(O-F)(l-t)

(8.24)

and the total after-tax earnings generated for the security holders becomes = O(l-t) + tF

(8.25)

Substituting F = rD from Equation (8.7), and defining WL as the total market value of the securities of the levered firm and L = D/WL as the degree of financial leverage in the firm's capital structure, the total market value, W L, is obtainable by capitalizing the total net earnings of the security holders as shown in Equation (8.25). The entity theory at this point argues that the portion of that earnings stream defined as 0(1— t), or what would be the net after-tax earnings if no debt capital were employed, should be capitalized at a capitalization rate applicable to an all-equity firm, or at an equity rate of p that would in that case be the same as /, the average cost of capital. The remaining portion of the total income stream in Equation (8.25), it is argued, should be capitalized at the rate of interest actually being paid on the debt. Making these substitutions, the value of the firm is definable as

172 II The neoclassical tradition Equation (8.26) permits the derivation of the average cost of capital of the levered firm, /*, after taking account of the tax effect. Rearrangement of the equation leads to ^

^

= i*

(8.27)

The tax adjustment thus effectively reduces the cost of capital to the firm (see Bierman and Hass, 1973, p. 170, note 7, for an alternative view of this tax adjustment effect). By virtue of the tax deductibility of interest payments on the firm's debt capital, the total market value of a levered firm will, on the given assumptions of an arbitrage activity in perfect financial markets, exceed the market value of a comparable unlevered firm by tWLL, representing the present capitalized value of the annual tax saving realized by the firm. Entity theory of the firm's capital structure and the equilibrium theory of financial asset markets In the foregoing analysis a number of references have been made to financial asset market equilibrium. It should be no surprise that the results of the Modigliani-Miller entity theory can be demonstrated to be consistent with the general equilibrium conditions of the financial asset market theory developed in Chapter 7. A close kinship is established by the assumptions of perfect financial markets and an externally given market rate of interest. The principal result of the financial asset pricing model can, in fact, be transformed into the equity cost function in Equation (8.23). But this, of course, confirms that this widely espoused functional description of the cost of equity capital is subject to the same criticisms and reservations as the equilibrium asset market theory from which it can be derived (see Hamada, 1969; Haley and Schall, 1973, Chs. 7, 8, to which the following argument is indebted). In the following development, we assume again that firm A is completely equity financed with a market value of VA, and the expected rate of return on its equity ownership is E(RA) = E(XA)/VA

(8.28)

In this expression XA refers to the annual stream of benefits resulting from holding the shares of common stock, here interpreted as a random variable. In the case of the levered firm B, which is otherwise comparable with A, its borrowing will be assumed to take place at a given rate of interest RF. Referring to its debt outstanding as DB and ignoring the firm's liability to taxes, the expected rate of return on the equity is RFDB

(8.29)

8 Cost of money capital: further analysis

173

We recall the principal result of the equilibrium theory of financial asset prices, stated in Equation (7.23) as

E(Rj)=RF+[E(RM)-RF] We now let /x=[£(# M )-/? F ]/Var(# M ), and rewrite Equation (8.30) in the form RJ, RM)

(8.31)

Expressing Equations (8.28) and (8.29) in corresponding notation and making use of Equation (8.31) yields ! ^ - = RF + nCov(RA, RM)

(8.32)

and E(XA) RFDB

-

M)

(8.33)

In Equations (8.32) and (8.33), the final terms on the right-hand sides may be written in the following equivalent forms:

Cov(J?A, RM) = C o v [ ^ > RM]

(8.34)

and

yB

(8.35)

Substituting these equivalences into Equations (8.32) and (8.33) provides E(RA) = RF + ^- Cov(X A , RM)

(8.36)

y

(8.37)

and ^wy V * A , RM)

It is known from previous development that the rate of return on the levered equity shares of firm B can be expected to exceed the rate of return on the unlevered equity shares of firm A. We therefore subtract E(RA) from E(RB), or Equation (8.36) from (8.37), to yield

174

II The neoclassical tradition E(RB)-E(RA) = \x Cov(XA, RM) - ± I ^

(8.38)

If, now, Equations (8.32) and (8.33) are solved for E(XA) and the results equated, it follows that VA[RF + n Cov(RA, RM)] = VB]^i

COV(RB,

RM)+RF

y

j (8.39)

In this expression, we can substitute the equivalences in Equations (8.34) and (8.35) to provide +^

COV(XA,flM)]= VB[^ COV(XA, RM)+RF

M ^ j

(8.40)

From Equation (8.40), it follows that (8.41) which implies that VA = V B + D B

(8.42)

The significant result in Equation (8.42) states a principal conclusion of the asset pricing model from which it has been derived. It is that when the full equilibrium conditions are satisfied, the total market value of the levered firm B, shown as the sum of the market values of its equity and debt capital on the right-hand side of Equation (8.42), will be equal to the market value of a comparable unlevered firm, as shown on the left-hand side of the equation. This is the same result as was established previously under the assumptions of a smoothly functioning arbitrage mechanism in a perfect financial market. The total market values of comparable levered and unlevered firms, according to this vision, must be equal and invariant with respect to their financing mix. Making use of this result, Equation (8.38) implies that E(RB)-E{RA) = fi Cov(XA, RM) ^ y

(8.43)

Inspection of Equation (8.36) indicates that in the generalized equilibrium the asset market model has in view, fi Cov(XA, RM) = [E(RA)-RF]VA

(8.44)

and Equation (8.44) may then be substituted into Equation (8.43) to yield -RF] ^

(8.45)

8 Cost of money capital: further analysis

175

Rearranged, Equation (8.45) provides E(RB) = E(RA) + [E(RA)-RF] ^

(8.46)

The conclusion in Equation (8.46) states that in full general equilibrium conditions the rate of return on the levered firm's equity will be equal to that on the unlevered firm's equity plus a risk premium. This risk premium is equal to the market's risk premium on the unlevered firm's equity [E(RA)—RF] multiplied by the financial leverage ratio in the levered firm. This statement can be compared with the previous conclusion derived from the ModiglianiMiller entity theory of capital costs and stated in Equation (8.23). It is reproduced for convenience of comparison: p = i + (i-r)DIV

(8.47)

By comparing Equations (8.46) and (8.47), it is seen that both bodies of analysis lead to the same conclusion. The equilibrium-theoretic analysis of a perfect money capital market provides the same result for the cost of equity capital as the entity theory of money capital costs under the assumption in the latter theory of a freely functioning market arbitrage mechanism. Growth of the firm and the cost of equity capital The value of a share of common stock has been defined up to this point as the present discounted value, or the present capitalized value, of the future expected earnings accruing to the owner of the stock. In our optimization model, the value of the equity has been stated as V=ir/p, where TT refers to the residual income accruing to the owners and p describes the owner's required rate of return or capitalization rate. This analysis, however, has been cast for the main part in a static mold. The different approaches to the money capital and valuation problem that we have discussed have also been developed from the same static perspective. We did observe, in the introductory discussion of the firm's economic position and performance statements in Chapter 2, that not all of the residual earnings of the firm may actually be paid to the owners in the form of dividends. Part may be retained and reinvested in the firm, and the market value of the ownership stock may increase as a result, as the future prospective income stream rises and reflects the earnings on the reinvested capital. The valuation formula we have used may be accommodated to the possibility of the growth of the firm that results from such reinvestment. Let us first restate the economic value of a share of common stock as the present discounted value of the future stream of cash flow benefits that the owner of the stock expects to receive. The value of a share of stock may be defined as

176

II The neoclassical tradition ^

^

-

-

(8 48)

-

-

where the series of terms in the summation extends to infinity. Each term in the series in Equation (8.48) expresses the present discounted value of a dividend payment, where Dt describes the dividend to be received in year t. The value of the share of stock can be stated alternatively as

lf

<849)

or, when time is taken as a continuous variable, P0 = f~Dte-ktdt

(8.50)

If the dividend remains constant each year, integration of Equation (8.50) yields the result Po = D/k

(8.51)

and by transposition it follows that the required rate of return on the stock, k, is equal to the dividend yield, or k = D/P0

(8.52)

This formulation for k can be regarded as the firm's cost of equity capital in the static nongrowth case. If, however, earnings are retained and reinvested, it is necessary to make an assumption, for valuation purposes, regarding the rate of growth of earnings that can be expected in the future. Let us assume, therefore, that the firm expects a growth rate of g percent per annum. If we assume also a constant dividend or payout ratio, the dividends will also grow at g percent per annum. Then, taking Do to represent the current or most recent dividend, the dividend expected in year t in the future will be describable as (8.53) if discontinuous growth is assumed, or Dt = Doegt

(8.54)

under assumptions of continuous growth. The present discounted value of the dividend stream then follows as (8.55) or, under assumptions of continuous growth and discounting, as

8 Cost of money capital: further analysis = %D0e-"-'»dt

177 (8.56)

Integration of Equation (8.56) provides the value of the share of stock under the specified growth rate asumptions: k-g

(8.57)

It follows by transposition of Equation (8.57) that the required rate of return, or the cost of equity capital, can be specified as k = D0/P0 + g

(8.58)

The cost of equity is now equal to the dividend yield on the common stock plus the assumed growth rate.6 Variable growth rates and the cost of equity capital It is unlikely, however, that a firm's earnings and dividends will grow at a constant rate forever. A firm may expect a supernormal growth rate of gs to be realized for, say, n years into the future, but after that the growth rate will fall to a normal rate of gn. Again we can determine the value of the share of common stock by taking the present discounted value of the expected stream of dividends. But now it is necessary to proceed in two stages. First, the present value of the first n years' dividends may be stated in the familiar way as

To this must be added the present discounted value of the dividends that are expected to be received subsequent to year n. These can be discounted back to the year n by applying the same valuation principle:

This expression will be recognized, however, as a perpetual dividend stream commencing at the beginning of year n + 1. Applying the formula for a perpetual growth stream derived in Equation (8.57), the valuation element in Equation (8.60) can be expressed as 6

The analysis of growth stock valuation was first developed by Williams (1965 [1938]) and rediscovered by Gordon and Shapiro (1956). The model presented in the text has become widely known as the Gordon model (see also Gordon, 1962, Ch. 4, and Vickers, 1966).

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II The neoclassical tradition

(8.61) PVn = - ^ k-gn Equation (8.61) provides the present value at year n of the stream of dividends that is expected to be received subsequent to that date. In order to obtain the present value now of that stream, the value at year n, PVn in the equation, must be further multiplied by the factor 1/(1 + k)n. Bringing these elements together, the value of the share of stock under the variable growth rate conditions assumed can be written as (8.62)

The required rate of return on the firm's equity can be estimated as the solution value of the discount factor, k, in Equation (8.62) that equates the righthand side with the present value, Po, or the current market price at which the share of stock can be acquired. In principle, any number of variations of growth rates could be assumed in this way, and an implicit discount factor could be calculated to provide an indication of the firm's cost of equity capital.

Comparison of equity capital costs From the various perspectives and the controversies related to the firm's cost of equity capital, we can summarize the following five points of view or alternative definitions. In each case, the symbol k is intended to refer to the cost of equity capital. i. Nongrowth firm (ignoring the implicit debt cost effect)

k = D/P [Equation (8.52)]

ii. Dividend growth at an assumed constant rate

k = D0/P0 + g [Equation (8.58)]

iii. Dividend growth at assumed variable rates

k = solution value in Equation (8.62)

iv. Equity capital required rate of k = i + (i — r)D/V [Equations (8.23) return implied by the financial and (8.47)] asset market equilibrium theory v. Full marginal cost of relaxing the money capital availability constraint

,_

dr dp dK dK [Equations (4.46) and (4.47)]

8 Cost of money capital: further analysis

179

It should be noted that the equity capital costs specified in Equations (i), (iv), and (v) make no allowance for growth, whereas those in Equations (i), (ii), and (iii) make no allowance for risk or uncertainty. For purposes of judging the economic worthwhileness of a proposed investment expenditure, the firm would be advised to estimate the cost of capital from more than one approach or set of assumptions in order to envisage a range within which the cost might fall.

CHAPTER 9

The investment expenditure project

The cost of money capital, understood in one or another of the ways examined in the preceding chapter, serves as a criterion of the economic worthwhileness of investing in the firm. The addition of money capital to the firm, or the retention within it of the money capital that presently exists and may be transferable to other economic uses, may depend, of course, on a number of considerations. Investment may be made in order to protect or increase market shares and in anticipation of economic growth, to effect cost reductions or take advantage of technological change, or to maximize the size of the firm because of the benefits that might thereby accrue to the management. But wisdom dictates, when account is taken of the long-run viability and economic position of the firm, that regard should be paid to the fact that money capital is a scarce resource and that a cost has to be paid for its employment. A balance should be struck between that cost and the economic values or benefits that the employment of money capital can generate. Investment decision criterion The basic notations relevant to money capital investment decisions were encountered in Chapter 3. The investment decision requires, in the terms summarized there, that the prospective increment of economic value be no less than the amount of marginal money capital it is proposed to employ. Alternatively, the investment criterion was stated in terms of the anticipated rate of profit, or the internal rate of return, that can be earned on the investment. In those terms, a decision to invest requires that the prospective rate of return should be no less than the cost of the money capital employed. Much depends, in the investment decision, on the proper formulation of both the prospective cash flow streams and the cost of capital used to ascertain their discounted value or to function in other ways as the decision criterion. In this chapter we shall look briefly at both these issues. But it will not be possible or necessary to take up the large number of technical questions or computational methods that abound in the business finance literature (see, for example, Archer, Choate, and Racette, 1983, Chs. 6-10, and Brealey and Myers, 1984, Chs. 7-12). 180

9 Investment expenditure project

181

Relevant cost of capital Consider the basic valuation Equation (9.1), where St refers to the periodic net incremental cash flow attributable to the investment project and m refers to the firm's cost of money capital. The other variables are as defined in Equation (3.7) of Chapter 3, and we ignore for the present the possible liquidation value of the project, Jn, at the end of its economic life:

V= £s,(l+m)-'

(9.1)

t

Our task now is to specify the appropriate magnitude of the money capital cost, m, in the equation. We can consider the following possibilities. First, if the firm is employing only equity or common stock capital, it is concerned only with the cost of equity capital at the margin of capital employment. That money capital cost may be estimated in one of the ways summarized in equations (i), (ii), or (iii) in the concluding section of Chapter 8. Second, it may be desired, in ways again that we have summarized in Chapter 8, to take more explicit account of the risks involved in money capital investment. Such risks may emanate from the forces that determine the instability in the firm's operating income stream (taking account of operational leverage) or from the manner in which the employment of debt capital induces a prospective instability in the residual equity income stream (taking account of financial leverage). The relevant cost of capital, then, will depend on two different possible analytical perspectives that might be adopted. First, if an analytical commitment is made to the apparatus of the general equilibrium theory, and if, as a result, it is imagined that the financial asset market reflects, at any time, conditions that closely approximate equilibrium, the cost of the equity portion of the money capital employed by the firm may be defined by the equilibrium required rate of return on the firm's equity capital. This was derived in Equation (7.24) as E(Rj)=RF + pj[E(RM)-RF]

(9.2)

This describes the rate of return that, at the point of observation and in the assumed asset market equilibrium conditions, investors require in order to induce them to hold the common stock or equity securities of theyth firm. It therefore specifies the rate of return that the firm must be able to generate on its equity capital if new investment and financing is undertaken. Only by doing so will it leave the equity holders as well off as they would be if no such investment and financing were to occur. If, however, the firm also employed debt capital or had a levered financial structure, it would need to adjust that equity capital cost in order to arrive at

182

II The neoclassical tradition

the rate of return that must be earned on its marginal investment project. The relevant cost of capital would in that case be approximated by taking a weighted average of the debt and equity capital costs. In doing so, the cost of debt capital would be weighted by the proportion of the total market value of the firm's capital securities accounted for by the debt capital, and the equity capital cost as described in Equation (9.2) would be weighted by the proportion of equity capital in its total capital employed. The rationale for making use of such a weighted average cost of capital under assumed market equilibrium conditions was examined at some length in Chapter 8. But such a weighted average cost of capital is, as we have said, at best an approximation to the relevant money capital cost involved. Care must be taken to consider whether the risk inherent in the proposed investment project is comparable with the overall risk of the firm into which it is being incorporated. If the risk of the new project is estimated to be the same as that of the existing firm, the existing weighted average cost of capital may be used directly as the relevant capital cost. But if, on the other hand, a more risky project is contemplated, it will be necessary to increase the required rate of return, and thereby the effective cost of capital, to take account of those higher risks. A rigorous adherence to the cost-of-capital concepts implicit in the equilibrium asset theory requires the conclusion that "the true cost of capital depends on the use to which the capital is put" (Brealey and Myers, 1984, p. 165). We recall at this point the concept, developed at length in Chapter 7 and reproduced in Equation (9.2), of the firm's beta coefficient. The logic of the equilibrium asset market theory requires the firm to estimate the beta coefficient of any contemplated investment project and to define a unique cost of capital for each project. In Equation (9.3), the cost of capital for the ith project is referred to as rif the project's beta coefficient as pit the expected rate of return on the market security portfolio as E(RM), in the same manner as in Chapter 7, and the risk-free rate of interest that also entered the development of Chapter 7 is referred to as RF: ri = RF + pi[E(RM)-RF]

(9.3)

In this manner, the asset market equilibrium theory takes account of the risk inherent in any contemplated investment project. Moreover, a further and most important conclusion, from the perspective of that theory, is arrived at from the same direction of argument. This can be observed briefly as follows. Given that separate project betas can thus be calculated, it can be shown that in a perfect financial asset market, such as is assumed in the equilibrium theory, where extensive opportunities exist for investment portfolio diversification in the manner we have examined there is no reason why a firm should diversify into different kinds of investment projects. Theoretically, separate

9 Investment expenditure project

183

investment projects could be established as separate firms, and ownership claims to them could be marketed in the financial asset market. Investors could then make their own diversification by combining such claims in optimum-security portfolios. This conclusion derives from what has become known as the "Value Additivity Principle." This states, in effect, that a firm that holds only the two assets A and B would find that the market value placed on the firm would be precisely the sum of the valuations placed by the market on the assets A and B separately. Values are additive, and the total market value of all such assets must therefore be independent of the firm that legally owns them. Diversification at the level of the firm is accordingly unnecessary and can, indeed, diminish the potential economic welfare of investors by reducing what would otherwise be opportunities for investor portfolio diversification (see Brealey and Myers, 1984, pp. 132-35, 408, 728-30). The logic of this line of argument is that in the last analysis no reason exists for the existence of firms at all. They can, on this view of things, be conceptualized simply as bundles of contracts or legal claims, and in the perfect world of the equilibrium theory the justification for the reality of firms must be found in decidedly nonfinancial reasons. In our search for the cost of capital that should be used in investment decisions, it might be preferred to view the question from an alternative, or a partial equilibrium, imperfect capital market perspective. In that case, the question arises as to whether, at the point at which the investment decision is being contemplated, the firm can be understood to have been brought to its optimum size and its optimum production and financing structures. If that should be so, then as exhibited in Chapter 8, there is reason to believe that the weighted average cost of capital can again be employed as the investment decision criterion. For the weighted average cost of capital is then equal to both the full marginal cost of debt and the full marginal cost of equity. It is equal, in other words, to the full marginal cost of relaxing the firm's money capital availability constraint. But a further question of risk remains. It is necessary to adjust the costs of debt and equity capital for an estimate of the ways in which those capital costs will be affected by any change in the overall risk exposure of the firm if the proposed investment is undertaken. There may be reason to believe, on the other hand, that the firm has not, prior to the new investment and financing, been brought to an optimum structural position. Its money capital cost decision criterion must then be of the kind we examined in Chapter 4 as relevant to such an infra-optimum stage. That capital cost was interpreted as the full marginal cost of relaxing the money capital availability constraint. In Equation (4.47) of Chapter 4, we specified both the full marginal cost of debt and the full marginal cost of equity. The former, for example, was stated as

184

II The neoclassical tradition

FMCD = r + D dr/dD + V dp/dD

(9.4)

In Chapter 8 also an example was given of how the cost of capital should be adjusted to take account of all of the potential effects on capital costs induced by changes in the financial leverage ratios in the firm's financing structure. Then in Equations (7.38) and (7.39) of Chapter 7 it was observed that a further adjustment of prospective capital costs may need to be made to take account of changes in the so-called business risk of the firm. This latter point was commented on further in Chapter 8. These observations on the cost of capital, whether they envisage the implications of the general equilibrium asset market theory or the partial equilibrium analysis of the firm, still need to be carefully adjusted to take account of the future possible rate of growth of the firm and the incremental risks that might be entailed. We have given some examples in Chapter 8 of the cost of equity capital for a growth firm, depending on whether growth is anticipated to occur at a constant or a variable rate. In such cases, and assuming again that the firm is understood to be expanding along an optimally structured growth path, a weighted average cost of capital may be calculated by incorporating the appropriate equity cost expression in the capital cost formula. We shall return to the question of the possible growth of the firm in Part III, where reference will be made to the use of internally generated funds, or retained earnings, as a source of money capital. Because retained earnings constitute a source of equity capital that is a substitute for external issues of new equity shares, the cost of external equity serves as an approximation to the cost of retained earnings. A possible difference in costs arises for a technical reason that need not detain us at this point. This follows from the fact that if earnings are distributed to the equity holders in the form of dividends, the recipients will have to pay tax on them at their effective income tax rates. But if earnings are retained, the equity holders will receive the benefit, if all develops according to plan and the retained earnings are invested effectively by the firm, in an appreciation in the market value of the shares of common stock. The equity holders may realize part of that capital appreciation by selling off a part of their equity holdings. In that event, they would be required to pay tax, not at the income tax rate, as in the case of dividends, but at a lower capital gains tax rate. This difference in potential tax liability may reduce the effective cost of retained earnings relative to the cost of new equity issues.

Investment project expected cash inflows Investment project valuation also requires a careful specification of the cash flows that the activity is expected to generate during its economic life. This is referred to in the St variable in Equation (9.1). Such cash flows should be

9 Investment expenditure project

185

adjusted to take account of anticipated inflation in input and output price levels. If future selling prices are expected to rise along with some general inflation factor, care should be taken to adjust future labor and other input costs by an appropriate amount also. A more extensive examination can be inspected in the business finance literature we have referred to. We shall conclude this chapter with some minimal comments on ways in which the risks in investment activities might be taken into account.

Probabilistically reducible risk and investment decisions Let us suppose that the elements of the cash inflow vector, St, are not uniquely definable but are regarded as random variables describable by subjectively assigned probability distributions. We can envisage a separate probability distribution for each cash flow element. The distributions may or may not be identical for different periods. We conceive, then, of the expected value and the standard deviation of each such cash flow element. Our problem now is that of finding the present discounted value of a time vector of expected cash flows and defining the degree of risk associated with them. Consider first the present valuation as indicated in Equation (9.5): E(V)= £ £ ( S , ) ( l + m ) - '

(9.5)

Since the economic value is now functionally dependent on a series of random variables, that economic value must itself be interpreted as a random variable. The expected value of the implicit probability distribution of the economic value is then definable as the weighted sum of the expected values of the cash flow components. The decision criterion may then be set as E(V) § C

(9.6)

But this leaves unexamined and unspecified the degree of risk involved in the investment. We need therefore to determine the variance or the standard deviation of the economic value of the project. This, however, will depend on what we assume regarding the correlation between the successive elements of the cash flow stream. We may consider three different possibilities. First, the cash flow elements may be assumed to be independent, or the covariance between every pair of cash flow elements may be zero. The variance we are searching for can then be interpreted simply as a weighted sum of the variances of the cash flow elements. In Equation (9.7), which reinterprets the expected economic value of the project, we have made use of the notion of the weighted sum of the expected values of the cash flows. We let (1 + m)~l = 1/(1 + m) = w, and the expected economic value of the project is then interpretable as

186

II The neoclassical tradition

Table 9.1 Conditional probability analysis of project cashflows Period 1

Period 2 Cash flow ($)

Conditional probability

Joint probability

0.25

50 100 200

0.40 0.40 0.20

0.10 0.10 0.05

200

0.50

100 200 400

0.20 0.60 0.20

0.10 0.30 0.10

400

0.25

200 400 700

0.20 0.40 0.40

0.05 0.10 0.10

Cash flow ($)

Probability

100

E(V)= Zw'EiSt) t=

(9.7)

I

In the case of independent cash flows, the variance of the economic value can be specified as Var(V) = 2 w2rVar(S,) t=

(9.8)

I

Second, the elements of the cash flow stream may be perfectly correlated. Again an analytical specification of the variance of the economic value is possible: (9.9) This case of perfect correlation of annual cash flow elements may be interpreted to mean that if an increase of 5 percent were envisaged in any given cash flow element, every other cash flow element would be expected to increase by the same percentage. Third, the analytical specifications of the variance in Equations (9.8) and (9.9) may not be possible because the cash flow elements may be neither independent nor perfectly correlated. The relation - 1 < p^ < 1 may apply to the correlation between the ith and theyth cash flow elements. An application of conditional probability arguments is then necessary. In Table 9.1, the first period's cash flow is described by a subjectively assigned probability distribution. Then the second period's cash flow is described, not by a uniquely specified probability distribution, but by as many

9 Investment expenditure project Table 9.2 Probability distribution Cash flow sequence (from Table 9 • D($)

187 of cash flow sequences

Period 1

Period 2

Probability of realizing cash flow sequence

100 100 100 200 200 200 400 400 400

50 100 200 100 200 400 200 400 700

0.10 0.10 0.05 0.10 0.30 0.10 0.05 0.10 0.10

2 = Too conditional probability distributions as there are possible outcomes in the first period. The same process can be continued for any number of periods. Any given period's cash flow will be described by as many conditional probability distributions as there are possible outcomes in the preceding period. In the example of Table 9.1, the third period's cash flow would be described by nine conditional probability distributions. The magnitudes in the final column of Table 9.1 are obtained by focusing on the different possible cash flow sequences. For example, the first possible such sequence is $100 in the first period followed by $50 in the second period. The joint probability of realizing this sequence, or 0.10, is given by the product of the probabilities attached to each of the elements in the sequence. In this way, it is possible to build up a probability distribution of cash flow sequences. We note that the probability magnitudes in the final column of Table 9.1 sum to unity, thus providing the probability distribution described in Table 9.2. It is not necessary to take the details further. If each possible cash flow sequence in Table 9.2 is discounted back to the present at a discount factor equal to the firm's cost of money capital, those discounted values may be set beside the probability magnitudes in the final column of Table 9.2 to provide a probability distribution of the economic value of the investment project. Then the expected value and the standard deviation of that distribution may be computed by familiar procedures. Those data will then provide estimates of the expected economic value of the project and the degree of risk inherent in it, and they may be used to establish decision criteria in the manner we have previously envisaged.

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II The neoclassical tradition

By the various procedures examined in this section it is possible to apply the probability calculus to estimate the expected values and risks in investment activities. Similar procedures could be used in connection with other distributional variables, or variables that are assumed for analytical purposes to be describable by subjectively assigned probability distributions, though the detailed applications would, of course, be different in different cases. For example, the rates of return on investment projects could be interpreted as random variables, or the rate of return on a firm's common stock could be similarly interpreted, as was done at some length in Chapter 7 in the discussion of the equilibrium theory of financial asset prices and yields. In all such cases, notably in the investment project valuation and decision problem, the attractiveness of the project and the choice decision can be further examined in ways that are now familiar. Given the estimate of the expected value and variance (or standard deviation) of the economic value, those data describing the project could be substituted into a utility function defined over the same two moments of the probability distribution in the manner we developed in Chapters 6 and 7, thereby providing a utility-theoretic approach to decision criteria. Alternatively, the investment decision criterion may be simply stated as in Equation (9.6), subject to the proviso that the estimated risk does not exceed a specified critical level. The risk measure may be stated in terms of the coefficient of variation of the economic value, or the relative standard deviation as described by the ratio of the standard deviation and the expected value. Other approaches to risk analysis in investment activities, such as sensitivity analysis, are widely used and are discussed in the business finance literature we have referred to.

PART III

Postclassical perspectives

CHAPTER 10

Neoclassicism and an alternative perspective The place of money capital in the theory of the firm is interdependent with the analysis of production, pricing, and investment. The preceding chapters of Part II have examined these issues from the perspective of received traditions in the theory. But that analysis, though it has shed considerable light on the financing problem and the specification and usefulness of the cost of money capital, was shackled by the heavy weight of the neoclassical assumptions that it carried. In this chapter we shall look briefly at some of the logical and methodological difficulties inherent in the neoclassical apparatus of thought. Our discussion will not be exhaustive, and it will be designed principally to prepare the way for the alternative perspectives we shall consider in the remaining chapters of this part. Production and factor employment Consider first the production problem of the firm. The neoclassical theory proceeds on the assumption that it is possible to specify a production function in such a way that there exists an infinite divisibility of factor inputs and a continuous substitutability between them. The implicit factor relations that exist by virtue of their substitutability are described in terms of the isoquants in the factor input plane. Any such isoquant describes a locus of factor combinations capable of producing a specified level of output. Or more technically, the isoquant describes the set of factor combinations that produce the same output when the factors are combined in their technically most efficient way. The slope of the isoquant, such as we employed in Chapter 4, exhibits the marginal rate of technical substitution between factors. Difficulty arises from this point of view when the capital factor is considered. In what respect, and with what analytical implications, it needs to be asked, is capital to be considered a factor of production? If capital is considered simply in money value terms, the well-known problem arises of defining capital, or at least of measuring its magnitude. For a value cannot be placed on capital until a rate of return, or a discount factor of the kind we have employed in the preceding chapters, has been calculated. When that is done, such a discount factor can be used to value capital by applying the discount factor to the prospective income streams that capital is expected to 191

192

III Postclassical perspectives

generate and thereby determine their present capitalized value. This then provides the implied value of the capital factor. But if capital as a factor of production cannot be valued until the exercise that searches for its optimum employment level has been completed and its rate of return has been determined, no meaning can attach to the concept of the marginal productivity of capital. No datum is therefore discoverable with which other factors' marginal productivities can be compared. No possibility exists, therefore, of defining a marginal rate of technical substitution between factors or the ratio of the marginal productivities of the factors. The latter, it has been seen, is, nevertheless, a central concept in the neoclassical production theory. In the model presented in Chapter 4 this difficulty was alleviated by distinguishing between real capital and money capital. Money capital, it was observed, was not to be regarded as a factor of production. Rather, it served as a constraint that defined the firm's command over factors of production and the other assets that were required in order to establish and maintain an effective production process. Real capital, on the other hand, was regarded as a factor of production. But the real capital factor in the production function was understood to be not the real capital assets that the firm acquired, such as machine tools, for example, but rather the flow of real capital asset services per period of time. In this sense, the production function was considered a flow - flow relation, associating flows of product output with the flows of factor services as inputs to production. This construction, with its emphasis on the flow of real factor services, avoids the circularity of what otherwise exists as the capital valuation problem. But even this procedure fails to come to grips adequately with the realities to which the problems and theory of the firm should be addressed. The preceding device retains the assumption that a continuous substitutability between factors exists. In doing so, it retains the possibility that varying returns to scale may also exist and may be reflected in ranges of decreasing returns and an upward sloping supply curve for the firm. In actual fact, large numbers of firms produce under conditions that do not permit them to exploit the possibilities of varying factor proportions and factor substitutability. Their production possibilities reflect fixed-factor combinations or fixed technological coefficients of production. This condition would appear to apply in large sections of the manufacturing sector where the relations between machines and the labor hour inputs necessary to operate them are technologically fixed. Of course, technological change or the adoption of new ways of doing things can alter those technological coefficients, and degrees of factor substitution may be able to be achieved in long-run structural decisions. It is precisely that recognition of the long run that justified our argument in Chapter 4 that an ab initio structural planning decision was relevant. But the realities to which the theory of the firm needs to be addressed frequently require, in the light of

10 Neoclassicism and an alternative perspective

193

fixed-factor proportions and constant technological production coefficients, the assumption of constant rather than varying returns to scale. In that case, the firm's average cost curve will be horizontal rather than upward sloping, and within normal operating ranges the supply curve will also be horizontal or perfectly elastic. Output markets and competitive conditions This implies that the neoclassical assumption of perfect competition in perfect markets may not be adequately descriptive of the firm's realities. If firms are not atomistic competitors in perfectly competitive markets, they will possess, in the general case, varying degrees of price-making power. Firms are not then primarily price takers. The selling price at which their products are offered will be determined by the cost structures that set the level of their horizontal supply curves. Variations in product demand will then most likely be reflected in variations in the level of output, or possibly in inventory accumulation or decumulation by the firms, rather than in changes in selling prices. A further principal difficulty of the neoclassical analysis, therefore, particularly in the form in which it establishes the notion of an economy wide general equilibrium, is its assumption that prices are everywhere determined by supply-and-demand conditions. Price formation, on the contrary, may be such that industrial firms exhibit production and output variations at fixed prices, at least within normal operating ranges, rather than the generally conceived price flexibility. Theory of utility Central to the neoclassical theory also is the notion of utility that we have examined at some length. The theory, it has been seen, has expanded the utility analysis from the more familiar consumer utility or indifference theory to the construction of investors' utility functions defined over risk and rateof-return characteristics. We labeled that extension of analysis stochastic utility, and we have observed that assumptions of differing degrees of risk aversion may be introduced into it. Powerful use has been made of the resulting apparatus within the neoclassical equilibrium theory. But a problem arises in this body of analysis if there is reason to believe that the utility functions might not be stable over time, or if the arguments over which utility functions are defined vary, or vary in their relative importance, as changes occur in other economic variables with which utility is brought into relation. This implies that the widespread analytical use of the indifference curves that are generated by utility functions may not be capable of serving the highly important function required of them in determining equilibrium situations. For

194

III Postclassical perspectives

that reason, it will be necessary to suggest in the final chapter of this part a way of coming to grips with economic decisions that does not make the same kind of reliance on the notion of static utility optimization. Logic of constrained optimization On the production side, the neoclassical theory visualizes the firm as deciding on its optimum production levels against the constraint of specified cost conditions. Or it might be viewed as deciding on minimum cost production levels against a production function constraint. On the consumer side, a similar analytical process is envisaged. The consumer or the consuming household is understood to optimize the choice of commodity combinations in such a way as to maximize attainable utility or satisfaction against an income or expenditure constraint. Neoclassical theory is therefore substantially a theory of the economic activity of decision-making units that maximize, or optimize, certain well-specified objective functions in the presence of definable constraints. Methodologically, it is a theory of constrained optimization or the discovery of constrained extrema of relevant maximand or minimand functions. We made use of this methodology in introducing into the firm's decision nexus in Chapter 4 a money capital availability constraint. This alerted us to the manner in which firms must take account, in interpreting and specifying their money capital costs, of the full effects that follow from the introduction of new money capital. The relations we adduced remain highly significant for the firm's money capital decision. But it would be a mistake to imagine, as we acknowledged at that earlier stage, that optimization decisions can proceed on the assumptions that all determinant functions are as precisely and uniquely defined as our initial argument might appear to posit. The general nature and directions of the causal relations might thereby be established. But care must be taken to observe the manner in which, in the ever-changing uniqueness of decision points in historic time, the determination of the relevant outcomes on the level of money capital employment also varies. Historic time, uncertainty, and knowledge The reintroduction of the realities of historic time brings into focus a further principal difficulty of the neoclassical theory. It is essentially a timeless theory. It is static and has not been able to accommodate effectively the realities of actual time and its unidirectional historic passing. To the extent that the future is considered, the uncertainties inherent in it have generally been abolished by interpreting future-dated variables as random variables and assuming that their values can be described by subjectively assigned probability distributions. By this sleight of hand, the problem of knowledge, or more particu-

10 Neoclassicism and an alternative perspective

195

larly of ignorance and unknowledge, has been assumed away. This means that the neoclassical theory has placed heavy demands on credibility by the manner in which its assumptions as to knowledge have been fitted into its scheme of perfect market and perfectly competitive analysis. Perfect market theory has generally carried along with it the assumption of perfect knowledge or perfect foresight. To the extent that assumptions of this kind have been modified by the theory, and to the extent that analyses of imperfectly competitive conditions have been proposed, the assumption has been retained that whereas precise knowledge of future-dated variables might not be available, at least the future could be corraled by positing the form of the probability distribution that described future possible events and outcomes. But such an assumption of the form of a probability distribution is an assumption of knowledge. The realities of uncertainty and of ignorance are thereby assumed away, and decision makers are assumed to be incapable, in an ultimate sense, of being surprised. General equilibrium postulates The neoclassical theory has made much of the notion that its conclusions regarding production and consumption optimization could be incorporated into a general equilibrium analysis. Having explained the determinants of consumer behavior and the behavioral decisions of producing firms, and assuming that at any time the total availability of resources in the economy is given, all decision makers could be understood to make optimizing decisions in such a way that all markets, for both product outputs and factor inputs, would clear at optimal prices and volumes. General market clearing is in this way taken to be symptomatic of, and descriptive of, the general equilibrium situation that guarantees a full employment of all available resources, including labor, and a maximum economic welfare. The unemployment problem is solved by assuming it away. The general equilibrium scheme provides full employment by definition. Or at least whatever unemployment exists is interpreted to be voluntary unemployment. All suppliers are on their supply curves. The possibility of involuntary unemployment is eliminated by the underlying assumption that suppliers of labor, for example, will have decided, along with all other decision makers, on their optimum supply. This will be determined by their utility functions that include as determinant arguments the supply of effort and the level of income received. In actual fact, not all markets clear continuously at optimum price and volume levels. Coherence in such a generalized sense might not occur. Market failures exist, and the imperfections and rigidities that potentially give rise to them need to be accommodated in any scheme of analysis of the market system.

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III Postclassical perspectives

An alternative theory of money capital investment For our present purposes, the considerations we have raised will influence our understanding of the problems the firm faces in deciding on its employment of money capital. In the remaining two chapters of this part we shall focus on two main issues. First, we shall suggest in the following chapter how a more relevant theory of the firm and its money capital investment problem may be constructed. This will imply an examination, from analytical perspectives somewhat different from those adopted up to this point, of the interdependence we have already emphasised, namely, that between the firm's production, pricing, investment, and financing decisions. In particular, we shall take account of the potentially high degree of concentration of outputs among small numbers of imperfectly competitive or oligopolistic firms rather than the widespread distribution of outputs among large numbers of perfectly competitive or atomistically competitive firms. Pricing and production policies will be related to such competitive structures. In the light of that, reasons will be adduced why incremental money capital investment in the firm will be required, and the conditions under which money capital may be obtained will be examined. In this connection, we shall be able to rely on the basic notions of the costs of money capital that we have already adduced, understanding that full adjustments to such capital costs will need to be made for the incidence of the risks introduced by the operational and financial leverage structures of the firm. We shall be able to make use also of the notion already examined of the rate of return that incremental investment in the firm might be expected to generate. But in that connection, an advance will be made beyond the basic money capital cost argument we have considered so far. We shall pay particular attention to the firm's use of internally generated funds, or retained earnings, as a source of money capital. It will be seen that the decision as to desirable money capital investment and the need to generate a sufficiently high level of internal cash flow to accommodate those investment requirements will be relevant to the interpretation of the firm's cost structure and the setting of its output price level. Uncertainty and potential surprise Finally, we shall examine in Chapter 12 the ways in which, in all of the decision situations we envisage, account may be taken of the real uncertainties that abound. Uncertainty will not be collapsed into what we have referred to as probabilistically reducible risk. We shall introduce in that connection the notion of a potential surprise function to replace the probability calculus on which, in the preceding chapters, the neoclassical apparatus has made such heavy reliance.

CHAPTER 11

Production and the place of money capital

An adequate theory of the interdependence between the firm's production, pricing, investment, and financing requires an examination of both its shortrun and long-run behavior. A model of the firm's production decision will not only reflect its selling price consistent with the chosen level of output but will also explain why that price is set at a level adequate to accomplish two objectives. First, net revenues after the deduction of operating costs must be sufficient to provide an acceptable or required rate of return on the money capital invested in the firm. This rate of return must be available after the deduction from revenues of the periodic allocation to a capital asset replacement fund if, in the manner we have emphasized, the firm is to maintain its capital intact. Second, the cash flow generated by the firm will bear a relation to its regular capital investment expenditures. Part of the money capital necessary to finance that investment will be obtained, as a policy decision, from internally generated cash flows. The level at which the firm's selling price is set, as a markup over costs, will therefore need to be large enough to generate those required investable funds. The remaining part of the capital budget will be financed by making new capital issues in the external capital market. The availability of money capital is brought into relation with the demand for investable funds by the money capital supply-and-demand curves in Figure 11.1. The horizontal axis of Figure 11.1 describes the investment expenditure, lu undertaken in a given time period t. The vertical axis describes, first, the cost of money capital, r, at which the requisite money capital funds are available and, second, the prospective internal rate of return on investment projects at the margin, or the marginal efficiency of investment, k. The curve labeled D describes the firm's demand for money capital. By using such analytical methods as those summarized in Equations (3.11) - (3.13), it is possible to calculate the expected true rate of profit, or the internal rate of return, that can be earned on each investment project contemplated by the firm. This true rate of profit will be the discount factor that makes the present discounted value of the project's expected cash inflows equal to the capital outlay on it. If the investment opportunities are ranked by order of their expected rates of return, the cumulative investment opportunities schedule will provide the marginal efficiency of investment relation, D, shown in Figure 11.1. The 197

198

III Postclassical perspectives r k

y

s

\ r — A.' * M

0

\

i i i

1

A

I*

i7

1

X

£>

1 1

/

Figure 11.1 larger the level of contemplated investment expenditure, the lower will be the expected rate of return on the marginal, or the last added, project. Consider now the supply-of-funds curve, or the marginal-cost-of-moneycapital curve, MS in Figure 11.1. Note that a kink appears at point A. The portion of the curve extending from the vertical axis to point A describes the supply of money capital funds from internally generated sources. That is shown in the figure as a positive function of the effective cost of funds, r. The AS portion of the supply curve describes the rate at which, and the money capital costs at which, investable funds are available from the external capital market. The cost of internally generated funds or retained earnings is shown in the figure to be lower than that of externally available capital. In the discussion of the investment project decision in Chapter 9, it was observed that the cost of retained earnings will approximate that of new equity issues, or the rate of return required by the equity holders. This followed from the fact that the retention of earnings is a substitute for new equity issues, as both are sources of equity capital. But the equity holders might enjoy a tax advantage from earnings retention compared with the distribution of dividends. The retention of earnings, provided the funds retained are invested effectively by the firm, can be expected to lead to an increase in the market value of the equity shares. An investor could realize part of that increased value by selling off part of his holding, thereby making himself subject to capital gains tax on the realized capital appreciation. That gains tax, however, will in general be lower than

11 Production and the place of money capital 199 the personal income tax that would have been paid on the receipt of dividends. This relative tax advantage of retained earnings might therefore reduce their effective cost marginally below the cost of new equity funds. Moreover, new equity issues carry further costs that are avoided in the use of retained earnings. A new issue of equity will frequently require the new shares to be sold at a fractionally lower price than the current market value of the existing shares, such an underpricing being necessary to guarantee the successful flotation of the issue. Also, a certain amount of underwriting costs will be incurred. No such costs exist in the case of retained earnings, and this, together with the tax advantage, suggests a lower effective cost of internally generated funds. The retained earnings portion of the supply-of-funds curve in Figure 11.1 has been shown as rising slightly. This reflects the possibility that increased earnings retention, implying a lower dividend distribution, may cause investors to fear that the risks in the firm's operation may be adversely affected as investment proceeds further. If that is so, the required rate of return on the equity will increase, leading to the possibility of a dilution of the value of the equity holders' investment position. We examined this kind of effect when we observed the possible impacts from the marginal use of debt capital. There is no a priori necessity for these incremental risk effects to occur in the earnings retention case. But the incremental income streams that the retained earnings will generate remain in the future; and the further in the future they are, the more risky they may appear to be. The distribution of earnings as dividends, on the other hand, provides the equity holders with immediate cash benefits. But any such risk effect depends on the quality of the investment opportunities in which the management invests its funds. It has been argued by exponents of the equilibrium asset market theory that in perfect capital markets investors will be indifferent between dividend distributions and retained earnings (see Brealey and Myers, 1984, p. 336f.). This so-called dividend irrelevance theorem is based on the assumption that a firm's capital expenditure program has been decided upon and that earnings retention and new capital issues are then two alternative ways of financing the given capital budget. Quite apart from whatever difficulties attach to the assumptions of the perfect capital market theory, investors who might be concerned about the incremental risks associated with the investment of retained earnings may visualize a somewhat different set of relations. Rather than contemplating the use of retained earnings and new equity issues as simply alternative ways of financing a given capital budget, they may be concerned also with the way in which earnings retention may lead to investment decisions by the firm's management that would not otherwise have been taken. The MA portion of the supply-of-funds curve in Figure 11.1 may also reflect an increase in the cost of internally generated funds that may occur if, as

200

III Postclassical perspectives

we shall examine below, the firm's selling price is raised in order to increase the availability of funds from residual cash flow sources (see Eichner, 1985, Ch. 3). This points to an examination of the relation between the firm's average cost markup, which translates those average costs into the firm's selling price, and the effective cost of changing that markup if investment or financial policies should make that desirable. The estimation of the marginal cost of external money capital, or the costs that are described in the AS portion of the supply curve in Figure 11.1, may proceed in the manner we indicated previously. Rather than relying on the assumptions of the equilibrium asset market theory, we may look directly to the marginal costs of debt and equity funds, or a combination of them, under imperfect capital market conditions. In other words, we rely again on what we exhibited earlier as the full marginal cost of relaxing the firm's money capital availability constraint. This may or may not, we have seen, be precisely measurable by the firm's weighted average cost of capital. If there is reason to believe that the firm is proceeding along an equilibrium growth path, and if, therefore, the firm raises its incremental finance in such a way as not to cause any change in its financing mix, and if it is maintaining production structures that provide the same degree of business risk, the weighted average cost of capital will be relevant and usable. But if, on the other hand, investment is being undertaken as a means of changing the firm's growth rate, perhaps by diversifying into other sectors of the economy where higher growth rates are attainable, a careful calculation will need to be made of the new overall risks that will be involved and the new implied costs of money capital. To the extent that firms do make new capital issues, the total supply of securities available in the financial asset market will be increased. The significance of this lies in the fact that in the exposition of the equilibrium asset market theory of Chapter 7 the assumption was made that the total supply of securities did not change. If firms do make new capital issues, the specification of the investors' asset opportunity set in the equilibrium asset market theory will be changed, and a different optimum market portfolio will be defined as a result. This will change the locus of the efficient boundary of the asset opportunity set, and it will alter, therefore, the characteristics of the Capital Market Line. This, in turn, will cause a change in the equilibrium market price of risk and a change in the effective costs of capital to all firms, or in the required rate of return on each firm's risky capital securities. In the situation envisaged in Figure 11.1, the firm would undertake an investment expenditure of 0/*, of which ON would be financed from internally generated cash flows and the balance, NI*9 would be financed by new capital security issues. Our task now is to develop a model of the firm in

201

11 Production and the place of money capital

MC

•AR

ATC = AVC + AFC AVC

60

80

100

% ERC

Figure 11.2 which the underlying determinants of these relations can be brought into clearer focus.1 Operating characteristics of the firm Let us imagine a firm whose production decisions are made on the basis of a fixed technological coefficient production relation. It will combine factor inputs in fixed and, in the short run, well-defined and unalterable proportions. We now depart from the earlier neoclassical assumptions of infinitely divisible and continuously substitutable factors of production. We may imagine that the firm is, in the expressive language of Eichner (1976), a "megacorp." It will conceivably arrange its production apparatus in such a way as to install a number of plant segments, each of which will operate with fixed factor combinations and will exhibit, as a result, a cost level dependent on its optimally attainable productivity. That will be determined by the technological character and efficiency of the particular vintage of capital equipment it contains. The operating characteristics of the firm can then be depicted in Figure 11.2. The horizontal axis of Figure 11.2 describes the Engineer Rated Capacity 1

The following sections are heavily indebted to Eichner (1976, 1985), whose work has profoundly significant implications, not only for the theory of the firm and its financing, but also for problems in macrodynamic theory and policy. Figures 11.2, 11.3, and 11.4 and the analysis to which they are related are essentially an interpretation of Eichner's work, and full acknowledgment is due to his important conceptual and terminological inventions. See also the related and highly insightful discussion in Wood (1975).

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III Postclassical perspectives

(ERC) of the firm's production apparatus. For production levels below 100 percent of ERC, the average variable cost per unit of output remains constant. This follows from the fact that the firm, as suggested, will be likely to maintain a number of plant segments, each of which will be brought into operation or taken out of operation as the firm desires to alter its total level of production. Each such production segment, however, will have been designed to operate at maximum efficiency, or at a lowest attainable average cost level, depending on its given vintage and technological characteristics. If it is called into operation, it will normally be operated at that optimum level of intensity. In this manner, the firm will be able to maintain a fairly constant level of overall average variable costs per unit of output. Or, perhaps, the average variable-cost curve depicted in Figure 11.2 might rise as the firm nears its 100 percent ERC, as older and less efficient plant segments are brought into operation in order to achieve the desired level of output. In the general case, a firm may aim to operate at a usual intensity of, say, 80 percent of its ERC as shown in the figure. Plant design may be such as to incorporate a target standard excess capacity. This, in turn, will provide the firm with the degree of production flexibility it may imagine it needs in order to satisfy the requirements of the output fluctuations it expects to encounter during normal business fluctuations. The average fixed cost per unit of output will, of course, decline as the level of output increases. When this is added to the average variable cost, the average total cost curve, ATC, is defined as in Figure 11.2. The marginal cost, which will again be constant (and equal to the average variable cost) at output levels below 100 percent of ERC for the reasons previously adduced, will rise if output is pushed beyond the 100 percent ERC level. The level at which the firm plans to operate can be referred to as its Standard Operating Ratio (SOR). It will be possible for the firm to determine at its planning stage the average fixed cost, average variable cost, and average total cost at the SOR level. The firm then faces the decision as to the price at which it should offer its output for sale. At that point, it must bear in mind that if it is going to maintain its share of its output market, it must plan to grow at the expected growth rate of the industry of which it is a member. This may be determined by estimates of the growth rate for the economy and by the likelihood that new entrants to the industry might succeed in reducing the firm's attainable market share. If, however, the firm is to retain its market share, and if it is to maintain the established degree of real capital intensity in its production arrangements, it will need to expand its capital stock each period at a rate equivalent to its expected overall rate of growth. This, then, will provide the firm with an estimate of its desired or target rate of incremental investment per period. Of course, other reasons may exist for incremental capital investment in

11 Production and the place of money capital

203

the firm. Investment may be made for purposes of achieving cost reductions, to install technologically more efficient or newer vintage capital equipment, or to diversify into new sectors of industry. The latter possibility introduces some potentially important analytical considerations. First, the firm may desire to diversify into a more rapidly expanding industry in order to increase its overall rate of growth. Or it may wish to diversify as a means of maintaining its growth rate if, for any reason, the attainable rates of growth in its currently established industry have slackened. But second, the suggestion that diversification might be desirable contradicts a significant conclusion of the equilibrium asset market theory. That theory implies that firms cannot accomplish anything positive for investors by undertaking diversification and that, indeed, the best interests of investors are served by leaving them to accomplish their own diversification by combining portfolio assets in the manner we have examined. In any event, the firm will decide, given the cost relations exhibited in Figure 11.2, the size of the markup it will add to its average total cost, as estimated at its SOR production level, to determine its selling price. The price it sets and announces to the market will then be defined as its average total cost multiplied by a markup percentage. We define the markup percentage as m and derive the relation between average total cost C and the selling price P: m = (P - QIC,

(11.1)

P = (1 + m)C

(11.2)

whence The apparent simplicity of this relation should not be allowed to conceal its significance. When production volumes are maintained at the SOR level on which the cost estimates and the selling price decisions were based, the firm will be realizing its target rate of return on the total money capital it employs. An alternative way of referring to its desired markup factor, then, is to say that it will mark up its average costs, and thereby set its selling price, in such a way as to realize a target rate of return on capital at its standard operating volume. The basic reason it does this, it should be borne in mind, is that it can thereby expect to realize also the internally generated cash flow it plans to use for financing its annual investment budget. If, in the course of business fluctuations, the demand for its product should increase, the firm's production can be increased above its SOR level without any change in its selling price. This, of course, is quite contrary to the vision of market price and volume changes implicit in the neoclassical view of industrial operations. In the present case, we have the prospect of price rigidity and volume flexibility. But at the same time, if the output volume is increased

204 III Postclassical perspectives in the manner envisaged, there will be an increase in the residual internally generated cash flow, or in the ACF magnitude (Average Surplus Cash Flow) in Figure 11.2. In that case, a larger amount of funds will be available for investment or for distribution to the equity holders in the form of dividends. Alternatively, depending on the level of incremental investment envisaged, the higher internally generated cash flows may diminish the need for reliance on new issues of securities in the external capital market. Moreover, if the production volume rises above the SOR level, the realized rate of return on the money capital invested in the firm will also rise above its previous target level. By the contrary argument, of course, if demand should decline and production levels fall below the SOR level, both the internally generated cash flow and the realized rate of return on capital will fall. In that event, the firm will have to decide whether it should reduce its capital budget or increase its demand on the external money capital market. Or it may, depending on whether it thought its diminished cash flow was a temporary phenomenon, increase its demands on the financial institutions for short-term loans, with a view to repaying them when the hoped for revival in demand and cash flows was realized. The extent to which that can occur, however, will depend on the creditworthiness of the firm in the short-term loan market, on the willingness of the financial institutions to supply the short-term accommodation, and possibly on the conduct of monetary policy that sets the tone and the overall availability of funds in the loan market. The firm may not, however, enjoy complete freedom to set its selling price in the manner we have suggested by marking up its average cost to realize a target rate of return on capital. It may be a price follower in its industry, not a price leader. Its demand curve, or its unit selling price curve, shown as AR in Figure 11.2, may be set at its existing level by the industry's price leader. In that event, the firm will decide on the standard operating volume at which it should plan to produce in order to achieve its target rate of return and cash flow. It follows that the average rate at which surplus internally generated cash flows become available differs from the marginal rate. The former is, as indicated, the difference between the selling price and the average total cost. But the marginal surplus cash flow is the difference between the selling price and the average variable cost. In conditions of rising demand, the average surplus cash flow will increase proportionately more rapidly than the marginal cash flow, with a contrary relation existing under conditions of reduced demand. We note finally the implications of this view of the firm's operations for the degree of risk to which it is exposed. If the firm is able to maintain its established share of the market or markets in which it sells its output, its principal risk will be that associated with demand fluctuations in the industry

11 Production and the place of money capital

205

market as a whole. Of course, there remains the risk that in the context of the growth of the economy and the industry, the firm may not be able to maintain the market share. The risk that new entrants to the industry may capture part of the firm's existing market share will be one of the factors that the firm takes into account in deciding whether or not it should make changes in the average cost markup by which its selling price is set. Moreover, there are also risks associated with fluctuations in the economy. These give rise to the fluctuations in demand that we considered above when examining their possible effects on the realized cash flows and rate of return on capital. In the event of such fluctuations, the degree of instability in the residual income stream earned for the owners will be determined by the overall degrees of what we referred to in Chapter 2 as operational leverage and financial leverage. Money capital employment decision The foregoing analysis implies that the incremental capital investment decision in the firm is integrated with its production and pricing decisions. The selling price at which the firm offers its output in the market will be set at that markup over cost that provides it with the internally generated cash flow it needs, along with the external financing it is prepared to undertake, to finance its capital expenditure. Let us suppose that in an established and ongoing situation the firm is generating precisely the amount of internal surplus cash flow it needs to implement its regular capital expenditure program. This, we may suppose, enables it to maintain its normally desired rate of growth, consistent with the expansion of its industry and with its established market share. We may consider, then, the possible ways in which the firm might finance a new set of investment opportunities in order to undertake a faster than previously planned expansion. How, in other words, should the incremental money capital be divided between larger internally generated cash flows and demands on the external capital market? Given the argument summarized in Figure 11.2, a higher average surplus cash flow could be generated by increasing the markup factor, say from m as in Equation (11.2) to m + Am. If the firm were able to operate at its previously established level of output, or at the same SOR level, the higher selling price implicit in the higher markup factor would, of course, generate higher internal cash flows. But whether the firm could maintain that production level will depend on the elasticity of demand for its output. It will actually depend on a number of such factors that we might inspect in the following way. Consider for this purpose Figure 11.3. In Figure 11.3(a) we have shown a relation between a possible increase in the markup factor m on the horizontal axis and the expected incremental surplus cash flow per period of time, Fit, that might result from such a decision.

206

III Postclassical perspectives AF/t

(AF/t),

Am

Figure 1 1 . 3

It is posited in the figure that increases in the markup factor will be likely to increase the surplus internal cash flow at a decreasing rate. There are a number of reasons why this might be so. First, the firm will have to consider, in its estimate of the outcome following a change in the markup factor, the extent to which the higher selling price will affect its position in the output market. It must take account, that is, of the substitution effects that might divert demand onto other competitive products, depending on the crosselasticities of demand that exist, as well as on the direct price elasticity of demand for the products of the industry of which the firm is a member. If the industry demand curve is inelastic, an increase in a selling price that is followed by the industry in general will increase the total revenue from product sales. A price increase under conditions of elastic demand, on the other hand,

11 Production and the place of money capital

207

will lead to a reduction of total sales revenues. The firm that is contemplating a possible increase in its markup factor will therefore make some estimates of the likely impact of these elasticity effects, particularly as they stem from commodity cross-elasticities of demand. Second, the firm will need to consider the likelihood that if it increases its markup and selling price, and if it increases its revenue by continuing to produce at its existing SOR production level, it will generate a larger rate of return on the capital employed in the firm. This higher rate of return might not be able to be realized permanently, of course, because the longer-run effects of its actions might offset any shorter-run benefits. The industry of which the firm is a member may be confronted in its immediate market by an inelastic demand curve, and firms in the industry may therefore be able to increase their revenues by increasing their markup factors. But in the longer run, the cross-elasticities of demand for substitute commodities could operate in such a way as to lower the industry's demand curve and thereby diminish every firm's revenue-generating prospects. But whereas attention is thus paid to the longer-run as well as the shorter-run effects, the prospect of a higher rate of return on capital may nevertheless attract new entrants to the industry. If the firm has reason to believe that such new entrants may emerge and in due course reduce its existing market share, that will also be taken into account in deciding on the wisdom of increasing the markup factor and the selling price. The extent to which such new entry problems may exist will depend, among other things, on the minimum size of the initial capital investment necessary to obtain a foothold in the industry as well as on other entry barriers such as access to input materials and complementary production resources. We may suppose, therefore, that the firm, in facing the decision whether it should increase its markup factor and selling price with a view to increasing its internal surplus cash flow for investment purposes, considers the following possible developments over time. It may estimate that in the short run, say the next few years, it may be able to increase its surplus internally generated cash flow by a certain amount if the markup is increased. But it estimates also that after that time it will begin to experience a reduced cash flow because of the kinds of considerations we have referred to. On the basis of projected data of this kind, it can perform a cost-benefit analysis, or an economic valuation or rate-of-return analysis of the kind we considered in Chapter 3 and other places. The anticipated ultimate reduction in the cash flow below what it would otherwise be will assume a calculable time shape. Similarly, the nearer term increase in cash flows generated in the firm will assume an estimated time shape. It is then a straightforward matter to compare the size and time shape of the ultimate cash flow reductions with the size and time shape of the im-

208

III Postclassical perspectives

AF/t

Figure 11.4

mediate cash flow benefits. The former can be regarded as the cost of realizing the latter. It is possible, therefore, to set these two estimated incremental cash flow streams, one positive and the other negative, against each other and to calculate the rate of discount that will reduce them both to the same present discounted value. The discount factor that establishes the equality will be an estimate of the effective cost of realizing the benefits defined in the relation between the two cash flow streams. The vertical axis of Figure 11.3(a) shows the annual equivalent incremental cash inflow that, it is anticipated, would result from different possible increases in the markup factor. In Figure 11.3(b), the vertical axis depicts the effective cost of realizing those incremental cash flows by setting against the increase in the markup factor that would give rise to them the effective cost, or the effective rate of discount as described in the preceding paragraph. Taking Figures 11.3(a) and (b) together, the firm now has, on the vertical axes, estimates of both the incremental annual cash flow and the effective rate of cost associated with any specified change in the markup factor. We may then refer to the effective rate of cost as the cost of internally generated money capital. The data on the vertical axes of these figures are brought together in Figure 11.4. Take first the curve OAC in Figure 11.4. This describes an effective supply curve of internally generated money capital funds, understanding that this analysis is intended to apply to incremental possible cash flows additional to the regular ongoing capital budget expenditures that are financed from exist-

11 Production and the place of money capital 209 ing markup and pricing relations and the established use of external capital market sources. The curve can be expected to assume the convexity illustrated in Figure 11.4 by virtue of its derivation from Figures 11.3(a) and (b), where the concavity of the incremental funds curve in Figure 11.3 (a) is married to the convexity of the effective-rate-of-cost curve in Figure 11.3(b). Inscribed in Figure 11.4 also is an incremental supply of money capital curve, HAS, describing the costs at which further amounts of capital may be obtained from the external capital market. This has been drawn with a positive slope to suggest that the capital market may impose larger required rates of return or higher effective costs of money capital on the firm as its capital expenditure continues to rise. Questions of risk that we have considered at length already come into view. If, as the level of capital financing increases, the firm makes increasing use of debt capital funds, the implications of the increased financial leverage can be expected to increase the cost of equity capital as well as raise the cost of debt, imparting an upward tilt to the cost of external capital funds curve. Much would depend also on the nature of the risks attaching to the firm's proposed incremental investments if it is investing as a means of gaining entry to a new industry, in which, conceivably, it does not have a history of experience and management. Moreover, the same effects as previously will again follow from the operational leverage and financial leverage that enter the firm's production and financing structures as a result of the incremental investment and financing. We introduce to Figure 11.4 also the firm's demand curve for incremental investable funds, understanding again that we are here describing the demand for funds over and above those provided for by the established markup and pricing policies and capital market relations. This demand-for-funds curve is juxtaposed with the effective supply-of-funds curve. The latter will be described by the internally generated cash flow curve extending from the origin to the point A, and thereafter by the AS portion of the external funds supply curve. The dotted portion of the internal funds supply curve, AC, is irrelevant because the cost of funds from that source has increased, beyond point A, above the external cost. Similarly, internally generated funds will be employed up to point A, as the cost of funds from internal sources is lower in that range than that of externally available money capital. The effective-supply-of-funds curve is therefore described by OAS in Figure 11.4. The optimum level of incremental investment, reading from the supply-and-demand cross at point B, will be (W. Of this amount, OAf will be obtained from internal sources, with the increase in the markup factor being set at the level that makes that possible, as derived from Figure 11.3(a). The balance of incremental funds, MN in Figure 11.4, will be obtained from the external capital market at an effective marginal cost of NB. Of course, the incremental demand-for-funds curve, D, in Figure 11.4 could

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III Postclassical perspectives

have crossed the effective supply curve OAS in the segment 0A. All of the necessary funds would then be obtained by making the change called for in the firm's markup factor and selling price, and no additional recourse would be made to the external capital market (see Eichner, 1976, 1985). It is conceivable that in the presence of temporarily surplus internally generated cash flows the firm could be a supplier of funds to the capital market. The firm's optimum-structure decisions The analysis we have developed in this chapter can usefully be compared with that in Chapter 4. In that earlier chapter we explored the ways in which, after refurbishing and expanding the widely accepted neoclassical theory, it was possible to conceive of a simultaneous solution to the firm's production, pricing, investment, and financing problems. The principal result of that analysis was the realization that the firm's optimum production structure was not independent of its optimum financing structure and that this carried along with it a fundamental message regarding the specification of the firm's effective cost of money capital. That was defined as the full marginal cost of relaxing the firm's money capital availability constraint. This latter concept can be carried over to the level of analysis contained in this chapter, as can also the basic notions of the valuation of income streams and the implicit rates of return on capital investment outlays. But some vital differences exist. In the earlier neoclassical analysis, no consideration was given to the possibility of earnings retention in the firm and the division of money capital sources between internally generated funds and financing obtained from the external capital market. That question was submerged in the light of the fact that the model of Chapter 4 was designed as a long-run planning model, which could, however, be reconsidered at any desired intervals of time. No division of the total equity funds into internal and external sources was made because, as we have seen, the costs of those funds are comparable, being close substitutes for one another. In the argument of this chapter, on the other hand, that division of fund sources has been brought into critical prominence. This has been possible for a further significant reason. In this chapter the firm has been regarded not so much as a static entity that structures itself once and for all at a point in time but as a developing structure that needs, if it is to maintain its place in a growing economy, to be continually expanding. It needs to reassess in each time period its optimum rates of capital investment and money capital usage, along with its most advisable financing sources, as well as reconsider its pricing policies, via its average cost markup policies, that make the realization of its overall goals possible. But one final point of similarity between the methodologies of Chapter 4 and the present chapter remains. In neither of these analyses is it intended to

11 Production and the place of money capital 211 leave the impression that the functional relations and the marginal values of outcomes, such as the comparison between the marginal efficiency of investment and the marginal effective cost of money capital in this chapter, can be specified with uniquely definable accuracy. In all of this work on the questions of the optimum structures and the decision problems of the firm, the realities of uncertainty and the relevance of the ignorance and the unknowledge that abounds impinge in ways we have not yet investigated directly on the making of management decisions. Our arguments have been designed to alert the analyst to the main directions of causation in some of the more critical relations that exist, or must be seen to exist, in the theory of the firm and its employment of money capital. In the following chapter we shall turn explicitly to the construction of a framework for decision making that addresses the uncertainties with which real historic time is inevitably pregnant.

CHAPTER 12

Uncertainty and decisions in the firm

The analytical issues of time and uncertainty have been relevant to many parts of our exposition. They throw an important light on many aspects of sequential decision making in the firm. In the traditional development of our subject, the assumptions that have been made in order to render the problems of time and uncertainty tractable have been related to the notion of equilibrium and equilibrium theorizing. In this chapter we shall reexamine a number of the interrelations that exist between these issues, and we shall bring into focus again a number of the questions that were raised in this connection in Chapter 1. The theory of the firm has all too often been cast in a timeless, static, certainty, or certainty-equivalent mold. The capitulation of economics to the analytical priorities of general equilibrium theory has left the theory of the firm preoccupied with the description of equilibrium states of affairs. This has become paramount so far as our principal object in this book is concerned, namely the analysis of the firm's employment of money capital. Money capital costs have been conceptualized by the theory as definable by the rates of return that investors require on risky assets when equilibrium conditions are satisfied in the financial asset markets. We have suggested, on the other hand, ways of envisioning the money capital problem from the perspective of partial equilibrium and imperfect market theory. Our move in that direction comes from the recognition that the general equilibrium theory, under its perfectly competitive assumptions, has no way of explaining how decisions are made in nonequilibrium situations. That, of course, is the only kind of situation in which decisions have to be made. If equilibrium obtains, the sea is calm and the ocean is flat and all is at rest. If equilibrium does not exist, the theory has no way of telling us how we can get into equilibrium. We know it only if we are there. The realities of time, moreover, bound us in a relative ignorance and shatter the comfortable epistemological security that assumptions of certainty and certainty-equivalents provide. Knowledge cannot be known before its time. But reality forces upon us the unavoidable moments of economic decision, and in them we form imaginative perceptions of the possibilities that the future contains. We inherit at our decision moments an accumulated knowledge and awareness and all the epistemological baggage that uniquely identifies us. 212

12 Uncertainty and decisions in the firm

213

We bring to those moments our particular complex of resources, endowments, skills, and capacities. We inherit a knowledge, or at least our private interpretation, of the fact situations that have structured the course of history that antedates our decision point. Our realization, or our imagination, of what fills out the bounded horizons of possibilities for the future is constrained by an awareness of the economic institutions that delimit our actions and color their outcomes. In such conditions our decisions are made. But in economic affairs we can never be sure of where we are going to be, or where, if we take certain actions, we shall arrive as a result. We can never know the end or the conclusion to which action will actually take us. We know only where we stand in the unique decision moments in the flux of time. Our task is that of interpreting our history, our present, and our tentative future in such a way as to make, in each unique situation, our best next move. How, then, can our decision moments have decisive meaning, and how are our decisions themselves determined? Historic time enters economic analysis when the passing of it influences our grasp of the uniqueness of the decision moments at which we stand. It casts its kaleidoscopic light on the imaginative perception of possibilities for the future that offers us the skein of conceivable outcomes and actions among which we choose. In the following sections of this chapter we shall allow these perceptions to throw further light on the theory of the firm and its employment of money capital. Possibility and economic decisions To begin an answer to these questions, we confront an issue that has partially intruded into our previous discussions. To what extent, we have asked, can the variables that influence decisions be properly understood as distributional variables, or subject, that is, to description by assigned probability distributions? This question raises acutely the meaning and treatment of uncertainty. The matter at issue is not simply that of uncertainty as distinct from risk, though in the fashion of Knight, Keynes, Shackle, and others, we have highlighted that distinction. The issue is grounded in the fact that the human perception of uncertainty is bound up with the perception of the passing and significance of time. Economic decisions are always decisions in time, temporally bounded and having temporal referents. The complexities of time cannot be elided by the conception that our vision of tomorrow can be safely delimited by our knowledge of yesterday. Let us imagine that a statement of probability is before us. This may be a statement that in a large number of conceivable experiments there is a wellattested reason to believe that in a specified proportion of the total number of outcomes a designatable result will emerge. In this sense, statistical probabil-

214 III Postclassical perspectives ities have meaning and applicability in economic affairs, in life insurance mortality tables, for example. In general, the experiments to whose outcomes probabilities are attached must be seriable, or replicable, and no aspect of uniqueness should surround the experiment to prohibit its recall and subsequent reproduction. On the other hand, the statement of probability may purport to be a description of the likelihood that in a single experiment, or as a result of a single and unique decision, a designated possible result will emerge. In that sense, the statement of probability has no meaning. The assignment of probabilities to future possible outcomes, either in the form of assumedly objective probability distributions or of subjectively assigned probability distributions, is actually an assumption of knowledge. It is this that makes the entire probability machinery inapplicable to decisions under uncertainty. For knowledge is the antithesis of uncertainty. It is the abolition of uncertainty (see Shackle, 1969, Ch. VII). The assignment of a probability distribution to possible outcomes involves the assertion that if the same event were to be repeated a large number of times, a designatable outcome would emerge in a prespecifiable proportion of instances. But such a replicability of experiments is precluded by the unique character of most economic decision situations. The taking of economic decisions frequently destroys forever the possibility of their being taken again. Decisions are in this sense "self-destructive." 1 For probabilities to be assigned over a set of possible outcomes, the assumption must be made that the set is complete and exhaustively descriptive of the possibilities without a residual hypothesis. As a homely example due to Shackle, imagine that of six candidates for a job one is, on all the grounds we can contemplate, the most qualified for the position. We accordingly assign probabilities of success to each of the six candidates with this imagination in mind. The probabilities assigned sum to unity and we have, supposedly, a bona fide probability distribution of the outcome. Suppose now that a seventh candidate is introduced, and that he appears to possess much the same qualifications as five of the six previous candidates. Now when we introduce the seventh individual to the candidate set, we assign a probability distribution over the seven possible outcomes. But we still have the necessary summation of probabilities to unity only by taking away part of the probabilities already assigned to one or more of the other six. If we try to capture the uncertainty The arguments in this and the following sections are heavily dependent on the work of Shackle, who has developed, in numerous books and articles over the last three decades, a critique of probability theorizing and has constructed an alternative potential surprise decision apparatus. These have been subject to extensive evaluation and criticism. See Shackle (1969, Ch. XI) for Shackle's evaluation of some of his critics, the recent extensive critique by Ford (1983), and Vickers (1978, Ch. 8). See also footnote 2.

12 Uncertainty and decisions in the firm

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by treating the outcome variable in this way as a distributional variable, we are making the assignment of probabilities dependent on the number of possible outcomes in view even though the very meaning of probability assignment implies the assumption that the distribution encompasses an exhaustive specification of the set of possible outcomes. We have supposed that the seventh individual possessed qualifications that were comparable with those of the other five and were again quite below those of our previous outstanding candidate. We therefore ask the telling question: How surprised would we now be if the outstanding candidate were awarded the job? Would we be less or more surprised than we would have been in the original situation? The introduction of the seventh candidate may make no difference at all to our contemplation of the outcome so far as the outstanding candidate is concerned. In the light of this, surely a means of handling uncertainty in decision-making situations needs to be devised that can take account of precisely this kind of possibility. This need arises because of what we have identified previously as the uniqueness of decision situations. Let us ask by way of further illustration whether it would have been meaningful for Napoleon to have assigned a probability distribution to the outcome of the battle of Waterloo. We may answer in the negative on the grounds that if he won the battle, there would be no need to repeat it, and if he lost the battle, it would not be possible to fight it again. In many economic situations, in investment decisions in the firm, for example, the decision events are unique in the same sense as was Napoleon's situation. The taking of the decision forever changes the decision environment and the decision nexus beyond recall. For these reasons, it appears necessary to replace the distributional analysis of probability by an analysis that focuses on the possibilities of outcome magnitudes of nondistributional variables. Potential-surprise function Let us approach the decision-making problem from the opposite side from that on which probability thought forms are imagined to be applicable. By asking the decision maker to describe the probabilities he would assign to certain specifiable outcomes, we are asking him to base his judgment on degrees of belief in conceivable results. Let us instead envisage a procedure that attaches degrees of disbelief to possible outcomes. We may imagine that when confronted with a range of possible outcomes, the decision maker specifies, on an ordinal scale, the degree of potential surprise he feels now, at the decision moment, he would experience in the future if various outcomes were to occur. By focusing on degrees of disbelief in the description of possible outcomes, our argument will be protected against any attempt to transform it

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III Postclassical perspectives

B + AX

Figure 12.1

into an argument in probability. The logic of the following argument is that the potential-surprise function is in no sense transmutable into a probability density function (see Katzner, 1986). Of course, the moment at which, in the future, the event that was previously in view actually occurs is different from the moment at which the decision is made on the basis of the contemplated potential surprise. No meaning can attach to the question of what surprise the decision maker would get in the future if a specified event occurred. He is concerned with the contemplation now of the potential surprise with which he contemplates the various possible outcomes of his decision. To give definiteness to these ideas, let us consider the relationship described in Figure 12.1. The horizontal axis measures the values of an outcome variable X, which we can assume for the present to be money values, thus providing a continuous scale. In different applications they may be, for example, the money values of cash flows generated by an investment activity or the rates of return on assets. Outcome values to the right of the zero point in Figure 12.1 are understood to be positive, and those to the left will be negative. The vertical axis of Figure 12.1 measures, on an ordinal scale defined on an arbitrary choice of origin and unit of measurement, the degree of potential surprise that the decision maker attaches to the various possible outcomes. To the possible outcome of OC, for example, is attached a potential surprise of CD. That magnitude will have been arrived at in answer to a question regarding the degree of disbelief with which the possibility of an outcome equal to

12 Uncertainty and decisions in the firm

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OC was envisaged, or the degree to which, in his present state and manner of seeing things, the individual imagines he would be surprised to see the value OC emerge. Proceeding in a similar fashion, it is possible to build up the entire curve y = y(x) describing the potential surprise attached to the entire range of possible outcomes (see Vickers, 1978). In contemplating the possible outcomes from a decision event, an individual will conceivably be able to specify a positive magnitude so large that he entertains an extremely high degree of disbelief in its possible occurrence. This may be so high that he does not regard the outcome to be possible at all. He regards it as perfectly /^possible. A maximum degree of potential surprise is thus attached to that outcome magnitude. We have labeled that maximum potential surprise as y in Figure 12.1. Drawing a horizontal line at that ordinate, we have shown the potential-surprise curve as asymptotic to it in the easterly direction. Similarly, the loss outcome segment of the potentialsurprise curve would be asymptotic to the maximum potential-surprise magnitude in the westerly direction. The rates at which the respective positive and negative segments of the potential-surprise curve approach the asymptote will not necessarily be the same. The one side is not necessarily the mirror image of the other. Everything depends on the determinants of the decision maker's imaginative view of things at the unique decision moment at which the various outcome possibilities are contemplated. Moreover, the potential-surprise curve should perhaps in certain instances be truncated in the loss direction. Suppose, for example, that the X values represented wealth outcomes from an investment activity and that the magnitude M on the horizontal axis represented the individual's total wealth and therefore his maximum sustainable loss. In that event, the loss segment of the curve to the left of the point N, which is vertically above the maximum sustainable loss M, has no meaning. For decisionmaking purposes, the curve should be truncated at that point. Consider now the segment AB on the X variable axis. Ignoring for the moment the remainder of the curve, imagine that in the decision event being contemplated the entire range of possible outcomes is described by a potential-surprise curve coinciding with the X axis throughout its length and terminating in A and B. It can then be said that the decision maker is attaching a zero potential surprise to all possible outcomes in this range. As he sees things, he would not be at all surprised if any outcome in the range AB occurred. For purposes of vocabulary, all outcomes in the range AB are conceived to be perfectly possible and have associated with them zero potential surprise. Continuing to imagine a function that terminates at A and B, the individual will, we suggest, focus his attention on only the best and the worst possible outcomes the project offers. In reacting to the ability of this project or eco-

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III Postclassical perspectives

nomic opportunity to arrest or grasp or command his attention, the individual will find his attention focused on the magnitudes A and B. These will be referred to as the focus elements, the focus loss and the focus gain of the project. For there is no point in an individual's being interested in the fact that an activity may bring a moderate fortune, say within the range of OB, if it is equally possible, or in this case perfectly possible, that it will bring a considerable fortune, OB. At the same time, there is no point in being concerned that the activity may result in moderate misfortune, say within the range of OA, if it is likewise perfectly possible that it will result in considerable misfortune, OA. "When many different things are all equally and perfectly possible," Shackle observes, "it is the brilliant and the black extremes which hold our thoughts" (1969, p. 118; see also Shackle, 1970, Ch. 5). In the case in hand, then, we can characterize the attention-arresting power of the project in terms of its focus elements A and B. In more complex and realistic cases, it is perfectly conceivable that within such a range "zero potential surprise can be assigned to each of an unlimited number of rival, mutually exclusive [outcome] hypotheses all at once; any number of suggested answers to a question . . . can all be regarded . . . as perfectly possible. The same is true of any other degree of potential surprise. . . . Potential surprise is completely nondistributional" (Shackle, 1969, p. 70). Suppose now that a project being contemplated differs from the one we have just considered in that its range of possible outcomes extends along the X axis to the right of B and also possibly to the left of A. Its potential-surprise curve can still be as shown in Figure 12.1, having an inner range of perfectly possible or zero potential surprise outcomes, but with the prospect also of other possible outcomes to which positive potential surprise is attached. Examine now the rightward segment of such a project's potential-surprise curve. A slightly higher outcome than B, say B + AX, may be conceived to be possible, but less than perfectly possible, and it may have associated with it a positive potential surprise indicated by the point W on the curve. As the level of possible outcome is higher, the degree of potential surprise will also become larger. Indeed, in view of the increasing degree of uncertainty surrounding the higher possible outcomes, and considering the fact that the possible outcomes become more shadowy in the decision maker's vision, the potential surprise may well increase for a time at an increasing rate. But for very high possible outcomes, for those, for example, in the range that is thought to be nearly impossible, the rate of increase in potential surprise will have slackened, and the potential-surprise curve will approach the asymptote y. It follows that the rightward segment of the potential-surprise curve is likely to assume the elongated S form shown in Figure 12.1. A comparable argument

12 Uncertainty and decisions in the firm

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can be applied to the S form of the leftward or loss segment of the potentialsurprise curve also. In the first of the projects we discussed, the project's power to arrest and command the decision maker's attention was encapsuled in the gain and loss focus values B and A. For the second project, as the contemplated possible outcome increases, say to B + AX, the attractiveness of this value, or its power to grasp or focus the decision maker's attention, may outweigh the nonzero degree of potential surprise associated with it. Thus the focus element of the project may shift from B to W. Then whereas the first project commanded attention by virtue of its focus elements A and B, the second project commands attention by virtue of its focus elements A and W. We can visualize in this way a cognitive and evaluative process in the mind of the decision maker whereby he marshalls together, with respect to a given project, both (i) the range of possible outcome elements associated with the project and (ii) the degree of uncertainty associated with each such element or subset of the total range of elements. If we assume that the decision maker is risk averse, to employ the parallel thought form we encountered in the theory of utility in Chapter 6, then he will be (1) attracted in a positive sense by higher possible money values of possible outcomes but (2) repelled by the higher degrees of potential surprise associated with them. A trade-off evolves in the decision maker's mind. He will want to choose projects or commit economic resources in such a way that if he is risk averse, he will have the prospect of realizing as high as possible a value outcome without the potential-surprise curve at such a point having risen too far above the X axis. Investment decision criterion We now recall that each potential-surprise curve that the decision maker confronts will refer to a specific investment project or other economic opportunity. If investment projects are in view, the decision maker will confront a nest or series of such potential-surprise curves, one for each project available for consideration. In respect to each potential-surprise curve, the decision maker must confront all possible outcomes extending over the range over which the curve is defined, but he must regard these not as possible occurrences in a series of repetitive processes but as rival outcomes. Only one of them can eventuate, determined by the unique conjunction of forces that give rise to it. Not only are all of the possible outcomes rivals in this sense, but the decision situation in which the decision maker stands is unique in the sense in which we have explained its uniqueness before. It is unique in that it occurs at a unique point in historic time, in the fact that the decision environment, existentially and

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III Postclassical perspectives

epistemologically, is therefore unique, and unique in the fact that the taking of the decision may destroy forever the possibility of its being taken, or even contemplated, again. But we can deduce a still firmer statement. As all of the possible outcomes of a given project over which its potential surprise function is defined are rival outcomes and only one of them can occur, no meaning at all could be attached to any kind of averaging of them, for no repetitive processes are available to give any rationale to such an averaging process. Potential surprise is completely nondistributional. No logical meaning can possibly attach to any such concept as the expected value or the variance of possible outcomes in such unique situations. Consider now a potential-surprise function of the kind described in Figure 12.1 associated with a specific investment project A. We now describe on the outcome axis the possible net present values of the project. We concentrate separately on its positive, or gain, and its negative, or loss, segments. We wish to isolate, in a manner consistent with the basic development to this point, what we called focus elements, or magnitudes on the value outcome axis on which the decision maker can concentrate his attention for purposes of evaluating the project. On the gain side, for example, such an element may be the value magnitude associated with points W or D on the potential-surprise curve in Figure 12.1. The focus element summarizes, as we observed in our development of the underlying potential-surprise function, the power of the investment project to arrest or command the decision maker's attention. Such an attention-arresting function, designated R in Equation (12.1), or what has been previously referred to as an attractiveness function (Vickers, 1978, p. 148), will be defined over the possible-outcome and potential-surprise magnitudes, x and y, respectively: R = R(x,y)

(12.1)

In Figure 12.2 are shown the isoattractiveness contours implicit in the attractiveness function described in Equation (12.1), together with a potentialsurprise curve. As will be indicated immediately, that focus element will exert the maximum attention-arresting power, or will provide the maximum attainable value of the attractiveness function, which establishes an equality between the slope of the isoattractiveness contour and the potential-surprise curve. The arrows in Figure 12.2 indicate the directions of increasing attracting power of possible focus points. Given the relation between the potential-surprise and outcome values, v = y(x), and recognizing that the potential-surprise function operates as a constraint on the possible values of the attractiveness function, or that the operative values of the latter function will be determined by coordinates that lie on the potential-surprise function, the attractiveness function becomes

12 Uncertainty and decisions in the firm

221

Figure 12.2

R = R[x, y(x)]

(12.2)

Taking R\ and R2 as the partial derivatives of the attractiveness function with respect to its first and second arguments, respectively, the attention-arresting power of a project is summarized in the following relations: R2 < 0

for 0 < y < y, x # 0

(12.3)

In determining the gain and loss focus elements, the decision maker will investigate the (JC, v) coordinates that generate the maximum value of the Rfunction in each of the gain and loss directions, understanding, as has been noted, that the relevant coordinates will lie on the potential-surprise function. The decision maker's subjective reaction.to possible outcomes and potential surprise summarized in Equations (12.1) and (12.3) therefore implies that the focus elements of the investment project under examination will be the points at which the subjective outcome-surprise trade-off implicit in Equation (12.1), dyldx = -RXIR2

(12.4)

is equal to the corresponding trade-off defined in the potential-surprise function that is relevant to the project. That is, dx

PS

dx

(12.5)

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III Postclassical perspectives

where the left-hand and right-hand sides of Equation (12.5) represent, respectively, the slope of the potential surprise function and the slope relation between the arguments of the attractiveness function. Such points occur at T and V in Figure 12.2, reflecting focus values of R and S on the x axis immediately below them. At those points, the highest attainable values of the /^-function will be registered. No possible (x, y) point on the potential-surprise function would generate a higher power of attraction or attention-arresting power as summarized in the attractiveness function. The point of maximum attainable R-value, therefore, will describe the (x, y) coordinates on the potentialsurprise function that serves as the focus element. Corresponding arguments thus lead to the specification of focus elements on both the loss, or negative, and the gain, or positive, segments of the potential-surprise function. The focus values describe the projects that the decision maker will subject to particularly careful scrutiny in the course of making his investment decision. He will be interested in those projects that promise high favorable outcomes with low potential surprise and/or large possible losses or unfavorable outcomes with low potential surprise. In his final project selection, the decision maker will be positively disposed to the high possibility, or low potential surprise, of large gains. Similarly, he will prefer not to invest in projects that hold out the prospect, with high possibility or low potential surprise, of large losses. In the example of the investment project, the loss could be visualized as negative net present values. We may therefore make use of the focus elements to establish an investment decision criterion in the following manner. In relation to the investment project, we now possess four pieces of information or input data for the decision event. These are (i) the focus element on the gain side, which we shall refer to in the following argument as FG, or, in the case of project A, as FGA; (ii) a comparable focus element on the loss side, FLA; (iii) a potential-surprise magnitude associated with the focus gain value, PS G A; and (iv) a potential surprise associated with the focus loss value, PS LA . The potential-surprise curve for project A with which we are working may not extend into, or may not extend significantly far into, the loss segment. Economic conditions may be generally so favorable that the firm can expect, with a high degree of confidence, to realize a positive outcome, or at the worst virtually break even, on the project. In such a case, the loss segment of the potential-surprise curve may be virtually vertical at or near the zero point on the outcome axis. What we are here referring to as the loss region may therefore include small positive values. The final investment decision leading to an acceptance or rejection of project A involves a subjective weighting in the mind of the decision maker of these four pieces of information. We may therefore conceptualize a Decision Index (DI) and stylize it as

12 Uncertainty and decisions in the firm DIA = /(FG A , PS GA , FLA, PSLA)

223 (12.6)

When the decision index value relevant to the project has been determined, the investment criterion can be visualized as DI ^ 6

(12.7)

where 6 describes a critical required level set by the firm. The project will be acceptable to the firm only if it has associated with it a Decision Index value no smaller than the prespecified critical level. This critical value may, of course, be zero. It will be set by the decision maker at a level that reflects his general aversion to uncertainty, as well as, in appropriate cases, the need to concentrate resources on investment projects that promise a rapid return cash inflow, or a low payback period. This may be desirable in situations, such as we encountered in the preceding chapter, where cash flows are highly desirable as a means of financing further investment outlays that facilitate the continued growth of the firm and promise highly attractive benefits in the longer run future. The general form of the Decision Index function in Equation (12.6) can be further specified as follows. Writing/, / = 1, . . ., 4, to refer to the partial derivatives, indicating the directions of change of the function value with respect to the four arguments in the order in which they are stated in Equation (12.6), we have /i>0,/ 2 <0,/ 3 <0,/ 4 >0

(12.8)

This means that high focus gains are desirable, but that high potential surprises associated with them are undesirable. On the other hand, high focus losses are undesirable, but that undesirability will be modified if a high potential surprise is associated with their possible occurrence. It is not possible to represent the four-dimensional Decision Index function geometrically, but an intuitive view of one of its planes can be seen in Figure 12.3. The figure shows the plane bounded by values of the first and the third arguments in the function described by Equation (12.6), given fixed values of the second and fourth arguments. Point A in the figure locates a pair of simultaneous observations from the focus gains and losses. We note that this point lies to the northwest of the curve 00, emanating from the origin of the plane. This curve might be understood as an indifference locus generated by the form of the Decision Index function, describing the manner in which, in the decision maker's mind, different combinations of possible gains and losses would contribute to the quantification of the Decision Index when the values of the remaining arguments are given. It is envisaged in Figure 12.3 that an indifference map may be deduced from the Decision Index function and drawn in a plane such as that indicated. The positive slope of the indifference curves

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III Postclassical perspectives FG

FG,

FL,

FL

Figure 12.3

follows from the posited signs on the partial derivatives in (12.8). The convexity in the curves suggests that as higher focus loss regions are contemplated, it will be necessary to offset these, for any value of the Decision Index, by higher focus gain values, the magnitude of which increases at an increasing rate as contemplated losses increase. Gain and loss combinations lying to the southeast of the 00 indifference curve, which we may refer to as the origin indifference curve, would then be less desirable than not committing money capital to any investment at all. So far as this plane is concerned, it would be desirable, for effective investment decisions, to choose available investment projects whose gain-loss coordinates, such as those represented by the point A, lie as far as possible to the northwest of the origin indifference curve, 00. In this way, it is possible to compare the desirability of different investment project opportunities and to rank them by the criterion of their Decision Index values. By this means, an adequately full attention can be paid to the characteristics of uniqueness surrounding each such project confronting the firm, to the uniqueness of the time point at which the decision is contemplated, and to the likely impact of a decision on the future structure and operating experience of the firm. It is not suggested, in other words, that the characteristics of uniqueness should be violated by the pretence, as is required by the alternative interpretation of the decision moment required by the probability calculus, that averages of potentially unique and rival outcomes can be meaning-

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fully calculated. Nor is it suggested that meaning can attach to the average extent of the dispersion of such potentially unique outcomes around any such average value. The notion of potential surprise from which the Decision Index is derived is, we have seen, nondistributional. Interpretation of economic value It was supposed, in the preceding example of the project investment decision criterion, that the outcome axis in Figure 12.1 over which the potentialsurprise function was defined could be taken to describe the net present value of the project. This implies that the firm first computes the possible net present values by employing the discounting procedures we outlined earlier and then describes the potential-surprise function over the values obtained. These actual computations would employ the basic valuation equation we took into account in the preceding chapter: V= X S , ( l + m ) - ' t=

(12.9)

I

In this equation, the discount factor is again taken as the effective marginal cost of obtaining money capital. We shall see below that the investment decision may be sensitive to the assumed possible values of the discount factor, and it will therefore be necessary for the firm to analyze its prospects and projects carefully in terms of a range of possible marginal costs of capital. Similarly, it will be necessary to assume ranges of values within which the periodic cash inflow, St in Equation (12.9), may fall. By such means, and by examining different possible combinations of periodic cash flows and money capital costs, it will be possible to build up a picture of the possible net economic values of the project over which the potential surprise function can be defined. Investment project selection can then proceed in the manner already indicated. But an alternative application of the decision methodology may be made. Let us focus initially on the successive periodic cash inflows and consider any one particular element of the series. We may specify the range of possible outcomes within which we assume the value of this St may occur and then describe a potential-surprise function in the familiar manner for that St. In this way we may focus, in the same manner as previously, on the focus gain and the focus loss of the rth period's cash inflow. Of course, it may again be the case that for specific periods, or perhaps for all periods, the potential-surprise curves defined over the cash flows might not extend into, or not significantly far into, the loss region at all. The cash flows may be confidently expected to be positive. By this means, we can specify the focus elements of each of the periodic

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elements of the cash flow stream. These focus values may be discounted back to the present by employing for this purpose the firm's marginal cost of money capital in the same manner as before. The successive periodic focus gains and losses should be discounted separately, in order to preserve the gain-loss distinctions we have worked with consistently in this approach to investment project evaluation. We thereby derive a present discounted value of focus gains that we can refer to as VG, or VGA in the case of project A. Similarly, we can derive a discounted value of the focus loss elements, referring to this as VL, or again as VLA. Referring to the magnitudes of the focus gain and loss elements of the possible rth period's cash flow as StG and StL, respectively, we have (12.10) and VL = 2 S , L ( l + m ) - '

(12.11)

At the same time as the focus elements of the successive periods' cash flows were specified to provide the basic data for the discounting or valuation procedure, the decision maker will have made subjective observations of the range of potential-surprise magnitudes associated with them. On this basis, it will be possible to specify also the potential-surprise magnitudes associated with the VG and VL values. In this way, the procedure will have provided the input values necessary to estimate the relevant value of the Decision Index. The only difference in the present case is that the first and the third arguments in the Decision Index function will now appear as the VG and VL values we have just derived. One further possible complication may, however, arise. It has to do with the possibility of what we shall refer to as investment project reversal. It can be envisaged briefly as follows. The possibility of project reversal Consider for this purpose the gain-loss plane of the Decision Index function as shown in Figure 12.3. In the present application of the project evaluation methodology, we may label the horizontal and vertical axes, respectively, as VL and VG. We again visualize the coordinates of project A in this gain-loss value plane. Suppose now that a lower cost of money capital is assumed for purposes of discounting the periodic cash flow elements. This will have the effect of increasing both the VG and the VL values associated with the project. The resulting effect on the project evaluation, so far as it is representable in the amended plane of Figure 12.3, may be to move the coordinate point from A to, say, M. If that should occur, the capitalized value of the gain outcomes

12 Uncertainty and decisions in the firm 227 has increased to a greater degree than that of the loss outcomes. The project's attractiveness has improved, as the coordinate point has moved further to the northwest of the origin indifference curve. On the other hand, it is conceivable that a reduction in the cost of money capital may cause a larger increase in the discounted values of the loss elements than in those of the gain elements. In that case, the coordinate point in Figure 12.3 may move from A to N. If this should happen, the project will fall from the northwest to the southeast of the origin indifference curve, and it will no longer be attractive to the firm. If the latter possibility should eventuate when a lower cost of money capital is assumed, the relation between investment expenditure and the cost of capital is running counter to that generally posited in neoclassical theory. Investment has become less, rather than more, attractive at lower costs of capital. The reason for this possibility lies in the different time shapes that may be assumed by the loss sequences and the gain sequences implicit in prospective cash flows. The possibility uncovers what became an intense ground of debate a decade ago under the name of "reswitching" and "capital reversal" in the theory of capital (see Shackle, 1969, pp. 238, 290; Harcourt, 1972). Utility reappraised Recalling the development of the preceding investment decision criterion, it would be an understandable inference to interpret the Decision Index as an implicit utility function defined over outcomes and potential surprise. Obviously, there is a formal similarity between the traditional utility function and the Decision Index, but the differences provide good ground for keeping the vocabulary quite distinct. The utility function, or the expected utility function defined in Chapter 6 and applied to the equilibrium theory of financial asset prices and yields in Chapter 7, was defined over the moments, principally the mean and the variance, of the subjectively assigned probability distribution of a random variable. But in the potential-surprise analysis of this chapter, the probability underpinning has been abandoned completely in favor of a nondistributional variable approach. There is therefore no warrant for speaking of expectations and associated dispersions in a mathematical-statistical sense, and such a procedure has been deprived of legitimacy. Any trade-off that arises in this new way of looking at things necessarily has a different connotation and is embedded in a very different specification of variables. The Decision Index, moreover, is tied uniquely to a specified decision moment in real historic time. At each point in time at which a decision is made, a new set of potential-surprise functions and a new Decision Index must be created, dislodging the old as events and attitudes are reshaped. Any notion

228

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of an intertemporally stable utility function comports best with a static view of decision situations and events, not with the dynamic evolution of affairs that belongs to the world of fact. Utility analysis and its associated vocabulary have meaningful reference only to the timeless and static constructs of the traditional theory that we have summarized in the preceding chapters. A final note on probability It may be imagined also that the potential-surprise curve can be regarded as essentially a probability distribution turned upside down and that all that has been said about potential surprise can be transformed into statements about probability. The suggestion may be encountered that the potential surprise attached to a possible outcome can be interpreted as unity minus the probability of its occurrence. But this would evince a failure to understand fully the decision apparatus we have proposed, and it would overlook some important and fundamental distinctions (see Shackle, 1969, Part II, Chs. 11, 12; Georgescu-Roegen, 1966, p. 260f. and Ch. 6; Katzner, 1986). It should be borne in mind that we are dealing here with potential surprise, and this requires us to bring into our analysis all that has been said regarding the uniqueness of the decision event and the decision maker's stance at a unique point in the flow of time. Potential surprise refers to an individual's assessment of what he imagines now, at the decision point, would be the surprise he would feel in the future if a possible outcome were to occur. There is no logical meaning in the comparison between surprise actually experienced at one point in time and the potential surprise with which a possible outcome was contemplated at an anterior point in time. Each such temporal point has its own uniqueness. But more particularly, the transformation of potential surprise, based on degrees of disbelief, into probability, which is related, on the contrary, to degrees of belief, is prohibited in a large number of economic-financial decisions by the fact that the occurrence of decisions at temporally separate and discrete dates makes it logically impossible to regard the decision events as repetitive or replicable. Consider the fact that our argument anchored the potential-surprise curve in the reality of perfect possibility. It would be well beyond our province to inquire into the psychological foundations that prepare an individual to take observable and describable attitudes to situations and possibilities and events. All that we have posited is that as the individual surveys the range of possible outcomes, he is able to say that there is a range of outcomes, a subset of the entire possibilities, such that he would not be at all surprised if any of them were to occur. Such an inner range need not be large. It may be quite small. The important thing is that, small or large, this is a range of outcomes that

12 Uncertainty and decisions in the

firm

229

the decision maker regards as perfectly possible and to which, therefore, he attaches zero potential surprise. To make probability transformations out of this would involve a gigantic leap. For what probability is to be taken as representing perfect possibility? Is it unity? But the analysis speaks of perfect possibility and not perfect certainty, for the whole tenor and motivation of the ideas is uncertainty. Moreover, there is nothing to prevent our saying that any number of outcomes are perfectly possible, and therefore assigning to them all a zero potential surprise. What is more, it is quite reasonable to assign perfect possibility to both a possible outcome and its contradictory. Inevitably this is involved when we invoke the uniqueness of the decision event, for from a given decision only one actual outcome can eventuate. What, then, would be the probability it might be wished to associate with our uncertainty concept of perfect possibility? In the light of what has been said, it cannot be unity. But then on what grounds is it possible to say that any designatable probability magnitude corresponds to perfect possibility? " Should we not rather say that there is no natural and self-justifying basis for a mapping of probability on possibility, and therefore, on potential surprise?" (Shackle, 1969, p. 71). This does not deny that the decision maker's judgments as to possibility and potential surprise may arise in part out of objectively given frequency ratios described in historically generated data. Yet, regardless of information, fashions, and even the conventional reliance on the likelihood that the outcomes of tomorrow will be like the events of yesterday, it does not follow that potential surprise can be mapped into probability. A hypothesis may, from a probability point of view, be rendered more improbable merely by a change in an individual's knowledge as time passes and introduces additional hypotheses, even though the first hypothesis is in itself quite unaffected by such new knowledge. Against all such changes as these, however, potential surprise is invariant. In connection with repetitive experiments, the idea of a probability distribution can find a haven. In such cases, the foundation postulate that financial theory deals with ' 'choices among probability distributions" would have meaning. But this, we have argued, is simply not the case in large numbers of economic-financial decisions. The latter involve preponderantly unique decisions under uncertainty, and the "unsureness" that abounds cannot invoke the probability methods that sweep uncertainty away. New knowledge does, however, appear as time passes. The experiences that time bequeaths and the actual surprises, delights, and disappointments, and regrets they administer shape our perceptions of things at succeeding decision moments. Our perceptions of the ranges of bounded possibilities may

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change, along with what history has taught us about the ways of anticipating different kinds of outcomes in different situations. Our ability to be surprised changes. Our understanding of our own potential surprise varies. The general structure of our expectation formation changes, though we realize that we never know more about the future. The future is forever unknown. It is unknowable. It is for that reason that we need an apparatus of thought with which to approach the task and responsibilities of decision making that acknowledges the realities of ignorance in which we inevitably stand. Further applications and the place of economic judgment The potential-surprise methodology has wider applications in the firm than simply to the capital investment expenditure decision we have examined. In the preceding chapter, the capital expenditure and the associated financing decisions were visualized in terms of a comparison between the cost of money capital and an expected rate of return on investment outlays. Though the discussion will not be extended at this stage, a potential-surprise function could equally well be described over a range of possible rate-of-return outcomes and the methodology could be implemented on lines parallel to the foregoing. The potential-surprise function may be described also over the possible rate-of-return outcomes or the possible economic values that can be realized from holding a portfolio of assets. The argument is not necessarily applicable only to the single-project or single-asset choice situation we have so far envisaged. The economics and the mechanics of asset diversification assumed a prominent place, of course, in the equilibrium asset market theory we examined in Part II. The most acute difficulty with it, however, is that it is grounded in the assumption that rates of return on assets can be properly treated as random variables describable by assigned probability distributions. We have discussed at some length the serious epistemological difficulties in imagining that the unknown and unknowable future can be corraled and its uncertainties defined away in this fashion. Moreover, the firms whose securities are assumed to generate the rates of return, as we observed at the beginning of our argument in Chapter 1, are not able to be treated as stable, unchanging, eventgenerating mechanisms. Firms change, grow, decay, and die. Their operating, financing, and general economic structures change over time. There does not therefore appear solid and secure reason to treat their periodic rates of return as random variables in a statistically manipulable sense. But individuals do hold, to varying degrees, portfolios of assets, and firms do invest in different asset forms, with a view to spreading risks and taking advantage of presumed or anticipated benefits of diversification. It might be agreed that the logic by which such diversification is grounded in probability assignments is defective. But there do remain reasons why investors diver-

12 Uncertainty and decisions in the firm

231

sify, such, for example, as the possession of specific information, hunches, guesses, or a willingness to rely on the positivist notion that yesterday's outcomes will in general be repeated again tomorrow or a readiness to imagine that historic distributions will continue to describe the future. Given that this is so, it is a straightforward matter to envisage the returns and benefits that any number of possible portfolios of assets might generate and to describe potential-surprise functions over them. From that point, a choice between portfolios may be contemplated in now familiar ways. Derived data descriptive of each of the portfolios may be brought into the Decision Index and their relative attractiveness compared. The strength of the analysis we have proposed lies to a significant degree in the place it accords to economic judgment in decision problems in the firm. Rather than assuming away the uncertainties for what they are and where they really exist, or reducing them to certainties or certainty-equivalents by stochastic reduction methods where the calculus of distributional variables cannot properly apply, the real-world decision maker's responsibility has been identified and provided with a logically robust framework of analysis for dealing with it. Judgments need not partake of the mathematical precision that might appear to inhere in the analysis as it has been presented. The main conclusion is that judgments involving comparisons between economic alternatives have to be made, and a quite simple and straightforward modus operandi for making them has been proposed.2 2

The foundations of decision making under conditions of uncertainty have become the subject of an extensive literature. On the proposals of Shackle, on whose scheme of analysis we have presented a variation in the foregoing, see the evaluation and critique in Ozga (1965, Ch. 7), Carter and Ford (1972), and Georgescu-Roegen (1966). In this literature, particular note might be taken of those proposals that, like our foregoing argument, suggest alternatives to probabilistic reductionism. See Ozga (1965, p. 263f.) for relevant comments on Savage's notion of minimum subjective loss (Savage, 1951), Hurwicz's index of pessimism and optimism (Hurwicz, 1972), Fellner's theory of safety margins (Fellner, 1960), and Roy's theory of minimum chance of disaster or what has become widely known as his safety first argument (Roy, 1952). Elton and Gruber (1984, p. 222f.) present a highly interesting reconciliation of Roy's safety first argument with the mean-variance theory of asset selection that we considered in Part II. This kinship stems from Roy's use of observed frequency distributions of rates of return and the application of Tchebycheff 's theorem on probabilities based upon them.

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Index

absolute risk aversion, 117-18 see also relative risk aversion accept-or-reject decision, 47, 222-3 accounts receivable, 27, 45, 61, 66 allocation, 108 Amey, L. R., 232 arbitrage, 149, 172 mechanism, 166-8, 170, 174-5 Arbitrage Pricing Theory, 149 Archer, S. H., 180 Arditti, F. D., 134 Arrow, K. J., 107, 118, 134 asset market theory, 133, 200, 230 see also financial asset market asset portfolio, 23, 103 attention arresting power, 219, 220-2 attractiveness function, 220-2 see also iso-attractiveness contours Average Surplus Cash Flow, 204 axioms of choice, 109, 119, 124 completeness, 119 complexity, 121 continuity, 120, 123 monotonicity, 121 substitutability, 120 transitivity, 119 balance sheet, 9-10, 22, 24, 66 analysis, 24-8, 37-8 homeostasis of, 27-8, 37 see also income statement Barges, A., 161 Baumol, W. J., 10, 14, 165 Bausor, R., 6, 17 belief, 215 Bernoulli, D., 108 beta coefficient, 146-9, 154, 155, 182 see also market beta Bicksler, J.,233 Bierman, H., 134, 172 Black, F., 148 book value, 24, 161 Boulding, K. E., 22, 23, 27-8, 37 break-even probability, 125, 126 Brealey, R., 10, 31, 134, 165, 180, 182, 183, 199

239

Brechling, F. P. R., 233 Buchanan, N. S., 14 capital, 22 budget, 197, 199, 204, 208 consumption, 24 see also depreciation expenditure, 205, 208 factor, 13, 23, 191-2 issues, 45, 199, 200 marginal productivity, 192 market, 50, 199,204,209-10 see also financial asset market mobility, 34-5 structure, 15, 169, 170, 172, 184 surplus, 26 see also debt capital, equity capital, money capital, real capital capital gains tax, 168, 184, 198 capitalization, 28 of equity income, 38-40, 6 0 - 1 , 74, 156, 175 rate, 60-1,74, 78, 80, 136, 164 capitalized value, 64, 175, 226-7 Capital Market Line, 142, 144, 150, 200 capital reversal, 227 Carter, C. F., 231 cash flow, 10, 28, 43-4 internally generated, 197, 204, 205, 207-8 marginal, 204, 208 sequences, 187 certainty-equivalents, 16, 212, 231 certificates of deposit, 11 Chamberlin, E., 4, 6 Choate, G. M., 180 choice, 7, 17, 108 see also axioms of choice criteria, 108 decision moment, 7 objects of, 109, 110 see also stochastic objects of choice theory of, 19, 108, 138 Coats, A. W.,233 Coddington, A., 23 coherence, 195

240

Index

common stock, 8, 13, 24-5, 39 see also equity capital competition, 5 atomistic, 193 imperfect, 5, 60 monopolistic, 5 perfect, 5, 193 completeness, 109, 119 complexity, 121 constrained optimization, 27, 73, 194 continuity, 120, 123, 124 Copeland, T. E., 12 correlation, 91-2, 95, 104-5, 185 coefficient, 92, 136, 186 cost, 60, 129, 202 average, 202, 204 marginal, 131, 204 cost-benefit analysis, 207 cost of capital, 10, 147, 148, 156-60, 209 see also money capital, real capital, weighted average cost of capital covariance, 88-91, 94-6, 131, 136, 185 matrix, 96-101 Davidson, P., 6, 10, 17, 18,23 debt capital, 12-13, 14, 31-3, 38, 67 full marginal cost, 75-6, 78, 80, 159, 160, 162, 183-4 marginal direct cost, 62, 75, 158 debt-to-equity ratio, 15, 29, 33, 61, 158, 165 optimum, 74 see also financial leverage, financing mix Decision Index, 20, 222-5, 226-7, 231 function, 223 decision nexus, 215 decision variables, 69 decreasing returns, 192 depreciation, 24, 26, 30, 38, 44, 64-5, 68, 74 accelerated, 30 fund, 30, 63-5 dilution, 76, 156-7, 159 diminishing marginal productivity, 54 direct marginal cost, 70 disbelief, 215 discount factor, 10, 44-7, 187, 208, 225 discounting, 42 distributional variables, 16, 188, 213, 231 see also non-distributional variables diversification, 88, 92, 153, 183, 203, 230 dividend irrelevance theorem, 199 dividends, 12, 14, 24, 184, 199 dividend yield, 177 durable factor capacity, 63 Durand, D., 165 economic judgment, 230 economic life, 10, 44, 46, 63-5

economic performance statement, 28-31, 37 economic position statement, 24-8, 37 economic value, 10, 38-40, 41, 43, 45, 50, 225 criterion, 45 maximization, 38-9, 54-5, 108 effective cost of capital, 200, 210 effective marginal cost, 69-70, 71-3, 225 Eichner, A. S., 20, 200, 201, 210 elasticity of demand, 206-7 cross, 206 price, 206 Elton, E. J., 31, 118, 134, 138, 147, 148, 231 endowment point, 51 Engineer Rated Capacity, 201-2 entity theory, 168-70, 172, 175 equilibrium, 4-5 conditions, 143-7, 165, 172, 174-5, 181, 212 see also general equilibrium, partial equilibrium equity capital, 14-15, 24, 38, 61, 67 cost of, 15,78, 153, 168, 171, 177-9 full marginal cost of, 79-80, 162, 183 excess capacity, 36, 202 expansion path, 73 expectations, 7 homogeneous, 135 perfect, 3, 16 factor costs, 28-9, 60 marginal, 72 factorial, 96 factors of production, 13, 29, 191 divisibility, 191 fixed proportions, 193, 201 substitutability, 191-2 variable proportions, 192 fair gamble, 117 false trading, 6, 135-6 Fama, E. F., 134, 165 Fellner, W.,231 financial asset market, 112, 151 equilibrium, 133-6, 143-6, 165, 172, 174-5 prices, 19, 133 theory of, 50, 133, 143, 182, 199, 230 see also Walrasian equilibrium theory financial asset portfolio, 19, 22 see also asset portfolio, optimum portfolio, portfolio theory financial firm, 22 see also non-financial firm financial intermediaries, 12, 22 financial leverage, 13, 31-4, 75-6, 79-80, 163-4, 170, 175, 209

Index financing, 19, 197, 210 decision, 20, 34, 36, 38, 74 mix, 15, 61, 164-71 see also capital structure, debt-to-equity ratio first-in-first-out, 30 fixed operating costs, 30, 37 flow-flow function, 9-10, 13, 192 flow variables, 22-4 focus elements, 218-9, 220, 222-4 gain, 221-4, 225-6 loss, 221-4, 225-6 Ford, J. L., 82, 107, 115, 119, 129, 134, 214,231 Friedman, M., 108 Friend, I., 233 full employment, 195 future-dated money, 41 Gabor, A., 14, 23 general equilibrium, 168, 172, 181, 195 theory, 5, 18, 50, 134, 153, 181, 212 see also Walrasian equilibrium Georgescu-Roegen, N., 228, 231 Gordon, M. J., 14, 177 Gordon model, 177 growth, 175-9, 184 rate, 176-8, 200, 202 supernormal, 177-8 Gruber, M. J., 31, 118, 134, 138, 147, 148, 231 Hahn, F. H., 18 Haley, C. W., 172 Hamada, R. S., 165, 172 Harcourt, G. C , 227 Harris, L., 23 Harris, S. E., 234 Hass, J. E., 134, 172 Henderson, J. M., 119 Herendeen, J. B., 60, 161 Hey, J. D.,82 Hicks, J. R., 5, 6, 17, 107, 134, 136 Hirshleifer, J., 17, 116 home-made leverage, 168 Hurwicz, L., 231 Hutchison, T. W., 3 - 4 , 6 , 17 ignorance, 3, 6, 11, 16, 195,211 imagination, 7, 213 imperfect markets, 168, 183, 200, 212 imputation rate, 70 income generating ability, 35 income statement, 22, 28-31, 37-8 see also balance sheet indifference curves, 110-11, 193, 223-4 see also origin indifference curve indifference map, 110, 116

241 indifference probability, 126-7 inflation, 185 interest burden, 6 0 - 1 , 158 interest imputation, 42 internal rate of return, 32, 47, 52, 166, 180, 197 interpersonal comparisons, 129 intertemporal consumption, 52-4 intertemporal production, 50-4 intertemporal valuation, 7, 9, 10, 14, 59, 133 inventories, 28, 45, 61, 66 investment, 7, 59, 191, 197, 210 budget, 203 criterion, 44-50, 180, 188, 219-27 see also economic value criterion, net present value criterion, rate of profit criterion decision, 19, 37-8, 44-50, 74 marginal efficiency, 10, 197, 211 non-normal project, 49 normal project, 49 involuntary unemployment, 195 iso-attractiveness contours, 220 isoproduct curve, 71-3 isoquant, 191 Jensen, M. C , 135 Kalecki, M., 12,23 Katzner, D. W., 216, 228 Keynes, J. M . , 3 , 6 , 11, 17, 23,213 Knight, F., 6, 17,213 knowledge, 3, 8, 15-16, 18, 195, 213 Koopmans, T. C , 15 Lachmann, L. ML, 3, 35 Lange, O., 14, 67 last-in-first-out, 30 Levy, H., 134 Lintner, J., 107, 134, 135 liquid assets, 7, 10-11 liquidation value, 44, 45 liquidity, 11, 66 Littlechild, S. C , 8 loanable funds, 23 Loasby, B. J., 6, 17,22 Lorie, J. H., 50 Luce, R. D.,235 maintenance and servicing costs, 63-5 Malkiel, B. G., 165 marginal cost, 131-2, 202 marginal imputed money capital cost, 70 marginal rate of substitution, 54, 110-11, 150, 153 marginal rate of technical substitution, 72, 191 marginal rate of transformation, 138, 153 marginal revenue, 62 product, 62, 69, 76

242

Index

marginal revenue (cont.) surplus, 76, 79 marginal value contribution, 77 marketability, 11 of assets, 11 market beta, 146 market clearing, 195 market failure, 195 market price of risk, 141, 143, 150, 153, 200 market share, 180, 202, 205, 207 Markowitz, H. M., 107, 134 mark-up, 20, 197, 200, 205, 210 factor, 203, 205-7, 210 Marris, R., 14 Marshall, A., 3, 4, 5,6, 59, 133 long run, 59, 133 short run, 59, 133 matrix algebra, 98 see also covariance matrix maturity date, 43 mean-variance theory, 88, 112, 114, 134, 150 megacorp, 20, 201 Miller, M., 10, 134, 164, 165, 169, 172, 175 Minsky, H. P., 12 Modigliani, F., 164, 165, 169, 172, 175 monetary policy, 204 money, 3, 18 function of, 18 money capital, 7, 9-10, 13-15, 19, 26-7, 35, 36, 59, 133 availability constraint, 14, 40, 67-77, 133, 156, 162 cost of, 19, 44-50, 60, 133, 149, 153-5, 156-79,181,198 effective marginal cost, 225 full marginal cost, 158, 183, 200, 210 intensity, 72, 73 internally generated, 208-9 marginal cost, 14, 45, 47, 200, 211, 225 marginal cost curve, 198 marginal efficiency, 71 marginal productivity, 68, 7 0 - 1 , 73, 75, 77-81, 158 market, 7, 12, 14, 61, 67, 168, 170 outlay, 63, 65 requirement coefficient, 61, 65-6, 69-72, 74, 76, 157 requirement function, 67 saturation, 67, 70, 72 sources of, 14, 36-7 suppliers of, 20, 39 traditional theory of, 168, 170 monopoly, 5 monotonicity, 121 Morgenstern, O., 108, 119, 129

Mossin, J., 17, 107, 116, 119, 134 multi-product firm, 36 Myers, S., 10, 31, 134, 165, 180, 182, 183, 199 net income, 29-30 net operating income, 29, 31, 162, 181 theory, 168-9 net present value criterion, 46 net worth, 25 New York Stock Exchange, 134 Nickell, S. J., 165 non-distributional variables, 215, 220, 225, 227 non-financial firm, 22 O'Driscoll, G. P., 8 operating flow costs, 37, 62-5, 69-70, 73, 76 operational leverage, 31, 154, 164, 196, 209 opportunity cost, 50 opportunity set, 109, 135, 137, 142, 149, 200 efficient boundary, 136-7, 141, 200 see also portfolio opportunity locus optimum financing structure, 210 optimum portfolio, 23, 141-3, 144, 148, 200 optimum production structure, 210 origin indifference curve, 224, 227 Orr, D., 10 outcomes, 216 inner range, 217, 228 perfectly possible, 217-18, 229 rival, 219 outcome-surprise trade-off, 221 Ozga, S. A., 82, 231 Page, A. N., 108 partial equilibrium, 168, 183, 212 Patinkin, D., 23 payoff period, 47, 223 payout ratio, 176 see also dividends Pearce, I. F., 14, 23 Penrose, E., 14 perfect capital markets, 50, 135, 151-2, 165, 168, 172, 182 perfect commodity markets, 129, 193 perfect foresight, 133, 195 Pigou, A. C.,4 planning, 160, 202, 210 ab initio, 35-6, 59, 67, 192 sequential, 35-6 planning problem, 160, 202 portfolio composition, 92 portfolio opportunity locus, 105, 107, 141-3, 153 portfolio theory, 133

Index possibility, 213-15 bounded, 229-30 perfect, 228 see also outcomes potential surprise, 8, 20, 215-19, 223, 228 curve, 217, 218-19 function, 196, 216, 220, 225, 230 maximum, 217, 218 zero, 217, 218, 229 Pratt, J. W., 118 preference ordering, 108-9, 128 representable, 109-10 preferred stock, 24-5 present-dated money, 41 present discounted value, 10, 28, 39, 44-55, 197 net, 46, 225 price leader, 204 price rigidity, 203 price takers, 193 pricing decision, 20, 191, 197, 210 probabilistic reductionism, 3, 16 probability calculus, 17, 20, 40, 188, 224 probability distributions, 7-8, 82-8, 213 a priori, 16-18 coefficient of variation, 85-6, 89, 188 conditional, 186-7 continuous, 86 density function, 86, 93 discrete, 86 dispersion, 84-5 expected value, 84 moments, 112, 114, 227 normal, 112 objective, 214 skewness, 87-8, 112, 114 standard deviation, 85-6, 89, 92, 136 statistical, 16, 213-14 subjectively assigned, 11, 15, 17, 112, 133, 135, 137, 185-8, 194, 214 variance, 85-6, 114 production, 7, 9, 19, 191, 197, 210 decision, 20, 34, 36-7, 74, 191, 197 function, 10, 36, 37, 60, 62, 69, 71, 1913 possibility frontier, 53-4 technological coefficients, 193, 201 transformation frontier, 52 product mix, 8 profit function, 68, 71 profit maximization, 38-9, 108 project reversal, 226-7 Quandt, R. E., 119 Racette, G., 180 Raiffa, H.,235

243 random variables, 15, 82, 114, 185 rate of interest, 7, 10, 12, 13, 23, 51, 61 risk-free, 134, 142, 146-7, 152, 182 rate of profit criterion, 47-50, 180, 197 rate of return, 8, 10, 112, 191 realized, 13, 204, 207 required, 34, 39, 45, 6 0 - 1 , 136, 138, 146, 153, 155, 175, 178, 197 see also capitalization of equity income, capitalization rate target, 203-4 real capital, 13, 22, 23, 27, 34-6, 59 intensity, 9, 29, 202 redemption value, 43 reference lottery, 125 refinancing, 75 reinvestment rate assumption, 48 relative risk aversion, 117-18 see also absolute risk aversion replicable experiments, 16, 229 representative firm, 4, 59 residual hypothesis, 214 residual income, 7 res witching, 227 retained earnings, 14, 29, 184, 196, 198-9 returns to scale, 192-3 revenue function, 62, 69 Ricardo, D.,4 Riley, J. G., 17, 116 risk, 11, 13, 15, 82-3 aversion, 39, 80, 88, 115-19, 125-6, 151-3, 171, 193,219 see also absolute risk aversion, relative risk aversion averter, 126 bankruptcy, 15, 82, 165 business, 154-5, 162-4, 200 exposure, 76, 168 interest rate, 11 lover, 126 own, 138 portfolio membership, 138 premium, 146, 149, 175 probabilistically reducible, 19, 40, 196 residual, 147 systematic, 147, 154 unsystematic, 147, 154 variability, 15, 82, 85-6, 154 see also market price of risk risk-adjusted discount rate, 133 risk class, 162, 164, 168 risk-free asset, 139-43, 150 risk-return trade-off, 11, 141, 143 Rizzo, M. J., 8 Robbins, L., 108 Robertson, D. H., 23

244

Index

Robichek, A. A., 165 Robinson, J., 4, 5, 6, 59 Ross, S. A., 149 Roy, A. D.,231 Sandmo, A., 129 Sarnat, M., 134 Savage, L. J., 50, 108,231 Schall, L. D., 172 Security Market Line, 147, 148 self-destructive decisions, 8, 214 selling price, 129 sensitivity analysis, 188 separation theorem, 50, 52, 135, 151-3 Shackle, G. L. S., 6, 8, 15, 16, 17, 18, 108, 213, 214, 218, 227, 228, 229, 231 Shapiro, E., 177 Sharpe, W. F., 107, 134, 138, 143 Smith, V. L., 14 Solomon, E., 50 Sraffa, P.,5 Standard Operating Ratio, 202, 203-4, 205 standard operating volume, 204 Stigler, G., 108 stochastic objects of choice, 129 stochastic reduction methods, 231 stock variables, 22-4 substitutability, 120 supply curve, 193 supply of funds curve, 198 Tang, A. M., 237 tax liability, 171 Tchebycheff's theorem, 231 technological change, 180, 192 time, 4-15, 18, 212,213 analytic, 6 real historic, 6, 7, 35, 194, 213, 227 time value of money, 42 Tobin function, 115, 118 Tobin, J., 107, 115, 134 transactions cost, 165, 168 transformation of assets, 27-8 transitivity, 109, 119-20 true rate of profit, 47-50 Turnovsky, S. J., 60, 161, 163 uncertainty, 3-4, 15-18, 59, 82, 133, 212 aversion, 223 decisions under, 214 residual, 15, 16 underpricing, 199 underwriting, 199 unique decisions, 8, 215, 217, 224, 228 uniqueness of the firm, 5 unknowledge, 195, 211 utility, 52, 108, 193 cardinal, 108, 129

construction of function, 122-4 expected, 113-15, 123, 129, 130, 227 exponential, 117 function, 54, 82, 87-8, 108-12, 122-9, 188, 227 intertemporal stability, 228 linear transformation, 128-9 marginal, 110-11, 113, 129, 131 maximization, 108, 111 mean-variance, 150, 153 monotonic transformation, 127 of expected return, 115 of risk, 115 ordinal, 108-9, 122, 127-9 risk-aversion, 115, 124, 130, 134 stochastic, 19, 82, 108, 114, 193 theory, 108, 135 uniqueness, 127-9 valuation line, 51-2, 54 Value Additivity Principle, 183 variable factors, 36 variable operating costs, 37, 202 variance, see probability distributions of weighted sum, 101-3 vectors, 97-9 algebra of, 98 multiplication, 99-100 Vickers, D., 6, 7, 8, 11, 14, 17, 23, 60, 63, 107, 134, 136, 142, 163, 177, 214, 220 vintage of capital, 202, 203 von Neumann, J., 108, 119, 129 Walras, L . , 5 , 6 Walrasian equilibrium, 4, 5, 18 theory of, 50 Waterloo, 215 wealth allocation, 150 wealth endowment, 135, 139 weighted average cost of capital, 160-2, 164-71, 182, 183,200 Weintraub, E. R., 4 Weintraub, S., 15-16 Weiss, N., 82, 134 welfare, 195 Westfield, F. M., 237 Weston, J. F., 12 Williams, J. B., 14, 177 Wiseman, J., 238 Wood, A., 201 working capital, 61, 66, 68, 74 requirement, 76 requirement function, 66, 68, 74 Worley, J. S., 237 zero-beta portfolio, 148-9 zero-risk portfolio, 106 zero potential surprise, 217, 218, 229